Is the date in your diary? The march for a people’s vote on 23 June marks the second anniversary of the Brexit referendum, that day of national meltdown and the harshest social rift of our lifetime. This week the Commons corridors are filled with intensive lobbying of MPs before next week’s pivotal Brexit votes. As amendments to the EU withdrawal bill are made in the Lords, whips demand maximum party tribalism. Meanwhile, rebels call on consciences to save the nation in its hour of peril. The vote on giving parliament a “meaningful say” on the deal was only won by four votes in December, so the decisions on staying in a customs union, a frictionless Irish border and joining the European Economic Area (EEA) hang by a whisker. But there’s a new spring in the step of the soft Brexiters and no-Brexiters. The crunch moment has come. A wind of change is blowing through parliament.

The cabinet’s shocking failure over two wasted years to resolve anything means that the baton passes to MPs. As rival cabinet groups debate impossible options that are not seen as viable by EU negotiators, nothing has changed, and still the clock ticks down to the 28 June summit in Brussels.

On this epoch-defining issue there is no government at the helm. Even Conservative party members say so; two-thirds told ConservativeHome they had lost confidence in their government’s handling of Brexit. With no one at the wheel, MPs must take back control.

Peter Kellner, a cautious pollster, says remain is now five percentage points ahead among voters, according to data from Delta. A majority would avoid the risk of a weaker economy in exchange for losing some sovereignty. By far the biggest gulf, greater than class or region, is between old and young voters: the retired and near-retired are still Brexiters, but by a margin of 65% to 35% working-age voters put the economy ahead of leaving the EU. That five-point remain lead is tight, but so was the four-point referendum victory that gave leavers such confidence that only the hardest Brexit is “the people’s will”.

To gauge the level of change, one need only note the growing fury of the Brexiters. Jumping up and down like raging Rumpelstiltskins, they sense the argument is slipping away. The leak of the Brexit department’s assessment of a no-deal scenario was splashed across the Sunday Times – that arch-Brexit paper. The second-worst scenario – not even their worst imagining of “armageddon” – warned that the port of Dover would collapse on day one, with food, petrol and medicines running out within days.

Cue apoplectic indignation from the Daily Mail, which calls the report “project fear on speed … tendentious rubbish”. It sneers at this report as emanating from Whitehall. But always remember that it was David Davis who commissioned it from the civil servants he appointed to his department. This was not even their worst prediction for the no-deal walkout favoured by the Rees-Moggites. Indeed, the worse the picture that emerges from every industry, the Bank of England, research scientists, the NHS, security and crime agencies – everywhere that matters – the more the Rees-Moggites rage with their fingers-in-the-ears, no-deal denial. They seem to be losing it, in every sense.

The Mail wills MPs to defeat the 15 Lords amendments “all designed to stall, dilute or reverse the referendum result”; all from the “unelected upper house stuffed with cronies and party hacks”. Hear that fear. “Unless Mrs May seizes back the initiative, they may succeed in wrecking the bill,” laments the Mail. And it looks increasingly possible that they will.

If so, expect MPs themselves to be branded “traitors”, “saboteurs”, “enemies of the people”. Expect democracy to be threatened by the Brexit fanatics. And know that this is the moment when each and every MP has to decide if they are merely subservient, or if – as Edmund Burke told the electors of Bristol – each, irrespective of party, has a higher duty. “His unbiased opinion, his mature judgment, his enlightened conscience, he ought not to sacrifice to you, to any man, or to any set of men living.”

What are the odds? It looks possible, maybe probable, that parliament will support a customs union and a frictionless Irish border, which means a close agreement on single market rules too. Sadly, Labour is deeply split on the Lords amendment for joining the EEA. Some 60 Labour MPs of the Caroline Flint, tough-on-immigration variety would vote against – even if the leadership swung to support it – despite enough Tory votes to win this too: 10 Tory MPs have spoken in favour of the EEA.

Even so, if the Commons votes for a customs union and a frictionless border, Brexit does take a far softer shape. And what will May do then? Breathe a secret sigh of relief, declare her hands are tied and pursue the least-damaging Brexit possible? The fanatics and the Mail will explode in fury, her party will be riven. What then? Does she turn to the people to vote on her deal? Do MPs insist that she does what they say?

Some remainers fret about a people’s vote but tellingly, it’s the leavers who fear that vote most. The shadow Brexit minister, Keir Starmer, says Labour is keeping its options open on the issue of a people’s vote. It all depends on the question: is it a soft or hard Brexit versus remain or leave? Or a three-way choice?

I will join the march on 23 June because if, as I hope, parliament votes for the softest Brexit, the claim that the people’s will is denied will need another popular vote to resolve it. And if parliament votes for a hard Brexit, then a vote is the last hope of saving us all. Above all, because I think the no-Brexit position could win. But please God, let there be no other referendum ever again: never forget the near-mortal damage done to Britain by an irrevocable vote on an unfathomably complex question. Instead, trust in general elections to throw the bastards out when they get things wrong.

• Polly Toynbee is a Guardian staff columnist