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It’s a fair bet that never, ever did the British establishment imagine that one day it would be resting its hopes on Jeremy Corbyn. For years, the likes of British Prime Minister David Cameron, Chancellor of the Exchequer George Osborne, governor of Bank of England Mark Carney, the heads of most FTSE (Financial Times Stock Exchange) companies and the masters of the City of London — to say nothing of Peter Mandelson and the entire New Labour aristocracy — would either have mocked Corbyn or struggled to place his name.

But now they’re relying on him. Last Thursday, the campaign for Britain to remain in the European Union (EU) exhaled collectively, relieved as the Labour leader weighed in at last — declaring himself to be in tune with the rest of his party, which, as he put it, is “overwhelmingly for staying in”.

The relief was intense because, without Labour voters, the referendum on June 23 is lost. With the campaign officially launched on Friday, and polls showing the contest too close to call, the numbers could not be starker. By one estimate, remain needs six million Tories and up to nine million votes from Labour supporters and others if Britain is to stay in the EU. And that’s presuming a general election-style turnout, when each party brings out its support more or less evenly. The great fear is of differential turnout, with outers more motivated than inners — so that while too many of the latter stay home the former stampede for the door marked “Brexit”.

The related anxiety is that the differential will be generational. The old vote. The young? Not so much. Younger voters are pro-EU, but that won’t matter if they sit out the referendum and let the old take the decision for them. One senior remain figure notes that June 23 coincides with Glastonbury: “That could be 150,000 of our voters too busy getting stoned in Somerset to turn up.”

That the in campaign is worrying about who’s headlining the Pyramid Stage is a sign of how nervy they are. Charles Grant, who heads the Centre for European Reform and is one of remain’s best-informed advocates, says that, as things stand, Britain is “probably heading for Brexit”, estimating leave’s chances at between 55 per cent and 60 per cent.

The sources of this concern are manifold. Start with the message. Remainers worry that there is a visceral simplicity to the leave case that is cutting through. I spoke to one United States pollster, entirely sympathetic to ‘in’, who admitted that when he looked at the key ‘out’ propositions put to British voters in focus groups, even he found himself nodding in agreement.

Would you rather laws were made by Britons or by foreigners? Would you prefer to give £12 billion (Dh62.62 billion) to the National Health Service or to the EU? Would you prefer that Britons or foreigners decide who can enter the United Kingdom? Grant admits that “the devil has the best tunes”, while the arguments for staying tend to be “complicated, boring and hard to explain”.

What’s more, leave has been able to swat aside remain’s best missiles with surprising ease. When ‘in’ says Europe needs to stay united against an assertive Russian President Vladimir Putin, ‘out’ has only to reply that France and Germany are bound to want an alliance with a post-exit Britain — and the argument seems to crumble. By the time the inners have explained the nature of deeper security cooperation and established alliances, the voter is yawning or has moved on.

And if the message is troubled, so too is the messenger. Until Corbyn’s speech, the ‘remain’ campaign was looking like a one-man show. Cameron has been on stage all but alone. Half his own MPs don’t back him and those who do have been conspicuously mute. British Home Secretary Theresa May has been all but invisible, and Osborne so sparing in his interventions, some wonder if he’s decided his only hope of becoming Tory leader rests on keeping his distance from remain.

And Cameron himself is weakened, thanks to the Panama Papers revelations about his finances and his hesitant, contorted response. The ‘in’ campaign drew up its battle plan assuming Cameron would be the same asset to them that he was for the Tories in 2015, sufficiently trusted to persuade the country that his assessment of the national interest was the right one. But that assumption now looks shaky. Britain is not immune to the anti-establishment mood spreading across Europe and the US and June 23 could offer the perfect outlet for it.

On top of all that are the very specific defects of the official ‘in’ campaign, Britain Stronger in Europe. Putting aside the confusion about whether it, or Downing Street, is in charge, even its allies despair that its top-down, old-fashioned reliance on corporate suits is a turnoff. They believe it has failed to engage wider civil society and to deploy the sort of voices younger voters might listen to — and that it lacks the killer instinct embodied by the ‘leave’ campaign’s standard-bearer Dominic Cummings.

Remainers have waited for the takedown that would expose the wild inconsistencies of Boris Johnson, for example, but worry that BSE (Britain Stronger in Europe) — a campaign with a name that sounds like a disease — lacks that ruthless appetite for the jugular. Even as Project Fear, it has failed: The picture it paints of a Brexited Britain is clearly not scary enough.

This is the context into which Jeremy Corbyn stepped last Thursday. The risk is that he might succumb to one of Ed Miliband’s great failings — and believe that, having given a single speech, he has done enough. He hasn’t. He has to campaign constantly and vigorously for in, between now and June 23. He should relish it — for this is a challenge for which he is unusually well suited.

His task is not to win over the entire country, where he might struggle, but Labour voters, where he should be strong — and, specifically, the young, where his admirers insist he is stronger still. He even has an organisation at his disposal, perfectly equipped for the task. He needs to unleash his grass-roots movement, Momentum, right away. Momentum can use the EU campaign as a demonstration exercise, proving its much-vaunted muscle by mobilising the Labour voters who will determine this referendum.

In this mission, Corbyn’s own backstory is an asset. With sincerity he can say that he empathises with the misgivings so many Britons have about the EU: He has them too. He can reiterate last week’s message that he has overcome his doubts, because he sees that the progressive cause — of protecting workers’ rights and combating climate change and tax avoidance — is best served as one of 28 nations rather than alone. And he can do all that in a language and demeanour that shows him to be as unspun and outside the establishment as Nigel Farage.

He can’t do it alone, of course. Hopefully, he’s now given the signal for Labour’s other big dogs to run — and it’s heartening to hear that an intervention is coming from Gordon Brown, eternally credible on this issue as the man who kept Britain out of the euro.

But such a burden falls chiefly on the leader. There are self-interested reasons for Corbyn to take it on. Here is one issue on which the whole Labour movement, unions and MPs alike, can unite behind him. If he succeeds in energising Labour voters, he can disprove those critics who insist he’s toxic at the ballot box. He can notch up a win. And, at the same time, Corbyn will do his country a great service, even shape its destiny. And not many predicted that.

— Guardian News & Media Ltd

Jonathan Freedland is the Guardian’s executive editor, Opinion.