Following two days in Salzburg, the British prime minister appeared to stick to her guns on the blueprint which is centred on a "common rulebook" approach, arguing that it was the only proposal that would avoid a hard Border in Ireland. "We need serious engagement on resolving the two main problems in the negotiations and we stand ready".
As the stalemate continued, the Irish protocol - a legal guarantee that would ensure no hard Border no matter the outcome of Brexit - again emerged as a sore point, with the European Union 27 rowing in behind Ireland's position that there can be no deal without first agreeing on the backstop.
Nigel Dodds, deputy leader of Mrs May's DUP minority government allies, said the events in Salzburg provide "more evidence of the unreasonable and inflexible approach of the EU" and said: "The UK Government must demonstrate a resolute determination not to be bullied".
He said: "Everybody shared the view that whilst there are positive elements in the Chequers proposal, the suggested framework for economic cooperation will not work, not least because it risks undermining the single market".
"If the political will is there on the other side I am confident we will reach a deal which is in the EU's interests as well as the UK's", she said. The first option would involve the United Kingdom staying in the European Economic Area and a customs union with the EU.
"As I have already said, that is unacceptable".
The Prime Minister took a direct swipe at Mr Tusk's comments and said: "Yesterday Donald Tusk said our proposals would undermine the single market".
Mrs May concluded her statement by saying: "The EU should be clear, I will not overturn the result of the referendum, nor will I break up my country. It would mean breaking up our country". He didn't explain how in any detail or make any counter-proposal.
"So Brexit tells us something - and it is with total respect for British sovereignty that I say this - it shows us that those who told us that they could easily do without Europe, that everything would go well, that it's easy and would gain them a lot of money are liars", he told reporters. "The UK expects the same".
Now that's expected to last until the end of 2020 but if there is no deal to avoid the hard border in Ireland and no political declaration outlining future relations then there will be no so-called transition period. His Remain-supporting Conservative colleague Anna Soubry compared May's Brexit plan to a dead parrot, while Iain Duncan Smith, a leading pro-Brexit MP, said that Chequers "clearly isn't going to fly".
Speaking to the media in Salzburg on Thursday, French President Emmanuel Macron said the staunch Brexit backers had deceived the British people, promising them a windfall of funds and a painless exit.