The UK and the EU remain a long way apart in talks on whether a revised Brexit deal can be agreed before the end of October.
Despite predictions that talks on a Brexit deal are on the brink of collapse, they struggle on.
After a series of meetings last week, the EU's chief negotiator, Michel Barnier, said "a lot of work needs to be done in the next few days" if any progress is to be made.
There is, however, a limit to what technical discussions in Brussels can achieve.
"We had serious detailed discussions," his UK counterpart, Brexit Secretary Steve Barclay, said "and we both want to see a deal".
This is about politics now - and time to get anything done and dusted by the end of the month has all but run out.
Technical teams are holding further talks and Prime Minister Boris Johnson has met European Council President Donald Tusk, among others, at the UN General Assembly in New York.
So, where have we got to?
Mr Johnson's overall approach is "let's just get this done".
The UK wants to rewrite the Protocol on Ireland and Northern Ireland that forms part of the proposed withdrawal agreement - the terms of divorce negotiated by the EU and the UK government under Theresa May.
His government is seeking to renegotiate the withdrawal deal agreed by his predecessor, Theresa May. But while it says progress is being made, no formal proposals have been tabled.
First and foremost, the UK now insists the backstop - the legal guarantee to avoid the return of a hard border in Ireland - has to go.
EU officials have previously suggested that the UK has been "dancing around the issues" during the talks in Brussels.
The UK plan to replace it would leave Northern Ireland inside the EU single market for all agricultural and industrial goods but outside the EU customs union.
The UK has now presented informal suggestions known as "non-papers", but they have been greeted with dismay by some EU officials, and on the substance it is clear that significant disagreements persist.
In effect that means a "light-touch" north-south border for customs (between Ireland and Northern Ireland) and an east-west border for regulations on goods (between Great Britain and Northern Ireland).
Irish backstop
Many businesses in Northern Ireland have already criticised the UK proposal as the worst of both worlds.
The UK is insisting that the protocol on Ireland in the withdrawal agreement has to be stripped back pretty radically - to remove the backstop. It is designed to keep the border between Ireland and Northern Ireland (which will be the only land border between the EU and the UK after Brexit) as open as it is now, under all future circumstances.
And the EU says it sympathises with companies on both sides of the Irish border that would suddenly face a far more complex and costly business environment to operate in.
The EU had previously said that the text of the withdrawal agreement could not be reopened.
But it still takes a more nuanced view, because it wants a deal done and doesn't want to be blamed if the whole process collapses in acrimony.
In a sign of some flexibility, that position has changed.
EU negotiators also appreciate the fact the UK has moved on the issue of regulations and has now proposed setting up a single zone, following EU rules, on the island of Ireland. That means checks on goods - especially food and agricultural produce - would have to take place within the UK (between Britain and Northern Ireland) instead.
It has now said it is willing to look at alternatives. But - in a statement issued after the meeting between Mr Barnier and Mr Barclay - it said any solution would have to be legally sound and meet "all of the objectives of the backstop".
But the EU has identified several other sticking points and concluded the UK proposal as currently drafted cannot form the basis for an overall deal.
That's a problem, because the UK appears to be asking not only for a different solution to the impasse, but for a different final outcome.
What about customs?
The backstop involves keeping many of the same regulations in Ireland and Northern Ireland in order to keep the border open, and to protect the integrity of the EU's single market.
This is the biggest problem.
It would also keep the whole of the UK in the EU's customs territory.
The UK wants customs checks to take place away from the Irish land border, using technology, exemptions for small businesses and trusted trader schemes.
But in a speech in Madrid on 19 September, Mr Barclay warned against seeking a "purist, identical result" which could only be achieved by never leaving parts of the single market and the customs union.
But that means it wants the EU to agree in advance, as part of a formal treaty commitment, there will never be any customs checks at the border itself, even before the EU knows exactly how the UK's customs proposals are going to work.
"The EU risks continuing to insist on a test that the UK cannot meet, and that the UK Parliament has rejected three times," he said.
The EU isn't willing to do that.
"We risk being trapped in a zero sum game, which will lead to zero outcomes, which I do not want."
It would also mean an international agreement known as the Common Transit Convention (CTC) would have to be renegotiated to avoid the need for documents to be checked at or near the border.
Mr Barclay also suggested that the details could be sorted out in a transition period after Brexit - a position that the EU has already rejected on many occasions.
And it would mean the EU would have to make special dispensations for the UK in Northern Ireland, exempting it from some of the customs rules set down in EU law. This, the EU believes, would threaten the integrity of its own economic structures - the customs union and the single market.
Any progress?
The UK is urging the EU to think more creatively. But overall, the EU still doesn't see how the Irish border can remain as open as it is now - certainly in the short and medium term - if Northern Ireland is no longer in the same customs territory.
So what progress, if any, has been made?
And it argues the UK plan relies on technology not yet deployed at any other border in the world.
An all-Ireland zone for food and animals, in which the Republic of Ireland and Northern Ireland would follow the same rules after Brexit, has been proposed by the UK, subject to approval in Northern Ireland.
What about consent?
A deal on the strict rules regulating this sector is vital because it makes up a significant chunk of trade across the Irish border.
This is the other big problem.
But it wouldn't cover everything - not even close.
The UK says Northern Ireland cannot be expected to accept rules and regulations over which it has no say, without first giving its consent.
And the UK has rejected any suggestion that such discussions could evolve into a backstop covering all aspects of trade for Northern Ireland only, rather than the current plan, which would keep the whole of the UK in the EU's customs territory.
But the method it proposes to gain that consent would - in effect - give one party, its allies in the Democratic Unionist Party (DUP), a veto over whether to create an all-Ireland regulation zone, and then another veto every four years on whether such a zone should remain in place.
"It's still a no to the Northern Irish backstop," one official said.
It's not clear whether the issue of consent could also extend in the future to any new customs rules - but the basis of the Good Friday Agreement that helped to bring peace to Northern Ireland was no single party could wield a veto over any aspect of the deal.
In other words, the UK wants Northern Ireland and Ireland to form separate customs and regulatory territories, with checks carried out in business premises, not on the border.
So, if a method could be found to seek the regular consent of all parties in Ireland, for post-Brexit trading arrangements, that could provide a way forward.
The UK team has talked of using increased surveillance on the movement of industrial goods, data sharing and tough penalties for infringement.
Although neither side would seek to call it this, it would be a form of time-limited backstop.
But EU officials have described the UK ideas as "conceptual" and "aspirational".
What about the rest of the withdrawal agreement?
"We want to keep this going," an EU source says. "But at some point the UK needs to give us a proposal. We can't negotiate without one."
The rest of the withdrawal agreement negotiated by the EU and Theresa May's government - including the financial settlement and the transition period, things heavily criticised in the past by Boris Johnson - would remain in place.
And there is obvious scepticism on the EU side about the ability of alternative arrangements to provide an overall solution.
So, even if a last-minute deal could be done - which looks unlikely before the end of October - the prime minister would have to accept parts of an agreement he has rejected in the past.
Alternative arrangements include:
What next?
"When they've been tested they haven't stood up to scrutiny, that's just the truth of it," Ireland's deputy Prime Minister Simon Coveney told the BBC.
The next few days are obviously critical.
On customs in particular, the UK proposals appear to be unacceptable. One leaked document said they would not be compatible with EU law.
Neither side wants to be seen as the one to walk away from a process of talks, not least because they know at some stage they would need to start talking again.
The essence of the EU argument? We can't replace a legal guarantee with a promise.
Sometimes in complex negotiations deals emerge at the 11th hour in the most unexpected of places.
Future relationship
"I think the deal is possible," the EU's chief negotiator Michel Barnier has said, "very difficult but possible."
But it's not just about the backstop.
But politics on all sides, and particularly in the UK, seems to have brought this process very close to an impasse.
Boris Johnson also wants a clearer path, in the political declaration on the future relationship, to what he calls a "best-in-class" Canada-style free-trade agreement with the EU.
It would involve the UK getting rid of many "level playing field" elements - promises agreed by Theresa May to stick close to EU rules on things such as subsidies for business, workers' rights and environmental rules.
That, though, could make it harder to reach an eventual agreement on a free-trade deal.
The EU is nervous about the UK, as a major trading partner right on its doorstep, undercutting its rules, and posing a significant competitive threat.
And the less likely it is a trade deal can be done relatively quickly in the future, the more likely it is the EU will stick rigidly to the terms of the backstop.
It's also worth noting that the UK's desire for a looser relationship involves not just economic issues, but defence and security too.
All of this exasperates the EU.
"The UK wants a less involved relationship," said one EU source close to the talks, "but it's not clear what that means in practice."
Time is tight
There is plenty of churn behind the scenes but little certainty about anything.
Philip Rycroft, who was until recently the permanent secretary at the Department for Exiting the European Union, has said it will be very difficult to get a deal done by mid-October.
"I think it is possible," he said, "but I don't see the other 26 countries ignoring the interests of Ireland… and time is crushingly tight."
"I'm not optimistic, and I'm not pessimistic," Michel Barnier said, quoting one of the EU's founding fathers, Jean Monnet. "I'm still determined."
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