With a late-night dash to Strasbourg yesterday, Theresa May secured a last-minute package of changes which she hopes to use to persuade MPs to back her Brexit deal this evening. A deal which – just two months ago – they overwhelmingly rejected. Has she achieved enough to change her critics’ minds? Perhaps in some cases, but certainly not all. But does that matter? Can she sneak her deal over the line with abstentions and some Labour rebels? We will find out later tonight in a crucial parliamentary debate.

I sometimes think that even if May had hiked up Mount Sinai and dragged down stone tablets from Moses himself setting out commitments that the backstop couldn’t become permanent, you would still have heard Eurosceptic MPs telling journalists that they would not support the Brexit deal. For some of her critics, no reassurance was ever going to work, and no compromise would be accepted. As Taylor Swift sang – haters gonna hate – and some critics will always oppose.

But there is also a biddable group of Conservative MPs who, although they were dissatisfied with the deal, and especially with the backstop, could be persuaded to vote for it. Many of them are worried about losing Brexit altogether, or seeing it diluted into an even softer version. They will have been worried to hear President Juncker insist yesterday that, if Brexit is delayed beyond May, the UK would have to elect MEPs, something that few in parliament want to see happen.

Will the reassurances be sufficient to assuage the concerns of these Conservative critics? A lot, perhaps everything, rides on what the attorney general, Geoffrey Cox, says when he publishes updated legal advice, later today. What the DUP decides will be critical too. If they declare they can live with the backstop as redefined, particularly given new binding commitments that it doesn’t cut across the 1998 Belfast Good Friday agreement, it seems likely that many erstwhile critics on the Conservative side could also soften.

On the other side of parliament, the Labour front bench continues its tantric dance towards (but seemingly never quite to) a second referendum. Having voted to support a referendum, promised to respect its result, and then whipped MPs to trigger article 50, the Labour party is now determined to vote against a Brexit deal almost come what may, primarily it often seems because it’s not a Labour Brexit (whatever that is). Keir Starmer, the shadow Brexit secretary – in a bizarre echo of May’s catchphrase – has tried his best to insist this morning that nothing has changed, when it obviously has.

Meanwhile, various backbench Labour MPs voted to trigger Brexit, decry a second referendum, abhor the idea of leaving with no deal, but also object to details of the current deal. Some of these MPs are happy with the entire divorce deal – the withdrawal agreement – but object to the future relationship as set out in the political declaration. Some would prefer a softer future relationship, inside the single market. Fair enough. But given these critics are willing to accept all the binding bits of the deal, and simply want to change the non-binding bits, it’s bizarre that they continue to insist they will vote the deal down.

Taken together as a package, the joint EU-UK legal instrument and the UK’s unilateral declaration shift the legal position for the UK. If Cox is able to update his advice, and argue that he no longer believes the UK could be indefinitely trapped in the backstop, then May could be inching closer towards finding a way to pass her deal. All of that may not happen tonight. But she will be hoping to narrow the margin of any defeat, and to find a path towards victory over the next few days.

Nothing that was secured yesterday provides a hard time-limit or a perfect unilateral exit mechanism from the backstop. But those options were always unlikely to be available so late in the negotiations. However, the withdrawal agreement already set out how the backstop could end or be replaced. What the new package does is clarify how that could operate in legally binding terms. It substantially improves the British situation, taking us to a place where both sides – the UK and EU – have what could be called suboptimal mutual reassurance. Neither side has a perfect position, but both can claim a strong hand which defends their interests.

A lot will come down to ladders or crosses. So late in the article 50 process, the options are narrowing. Will MPs think the concessions won yesterday are sufficient to provide them with a ladder to climb down? Or will they want to martyrise themselves to prove their pursuit of their perfect Brexit outcomes? We will find out soon enough.

• Henry Newman is the director of Open Europe