Labour’s fate – and with it that of the country – now rests with the young: not just the fresh-faced undergraduates of Lancaster and Cardiff but the sleep-deprived 35-year-old mums and dads of Hastings and Pembrokeshire. Just as at this stage in the 2017 election – and no, the past is no automatic predictor – Labour’s polling ranges from not good to very bad. But every scrap of evidence underlines that Labour’s prospects depend on whether Britain’s young apprentices, undergraduates and doting new parents turn up to the polling stations in the first place.

Ever since Labour surged from its pre-2017 election nadir, when it languished in the mid-20s in the polls, whether or not there was a “youthquake” has been debated and contested. But the latest research does indeed suggest that an increase in younger turnout played a key role. According to YouGov post-election analysis, an astonishing two-thirds of 18- and 19-year-olds opted for Labour, but the swing towards Jeremy Corbyn’s party among those in their 30s – 55% of whom plumped for Labour – played a decisive role.

It is fitting political karma that the Tory failure to address the housing and cost-of-living crises in London in particular have detonated in the party’s face. Unable to afford to raise children in the capital, young families are driven to Tory marginal seats, and have been turning them red. They’re known as DFLs in many of these places, “down from London”. In Hastings and Rye, they are playfully called “Filth” – “Failed in London, tried Hastings”. In this constituency, they helped bring the Tory majority down from approaching 5,000 in 2015 to just 346 two years later. They care about job security, affording a decent home, childcare and being paid properly; they were raised to expect that life would always get better, are affronted by their stagnating living standards, and worry about their children’s future in a turmoil-ridden world. They tend to have progressive social views, and they are disturbed by the US-style conservative culture war being waged by the Tory right. The climate emergency unnerves many of them.

But will the young vote? The worst poll for Labour, conducted by Kantar, suggests a horrifying 18-point Tory lead – though, eerily, its poll at this stage of the 2017 election suggested the same gap. Its findings are weighted by turnout: it claims that over three-quarters of Tory voters are certain to vote, but that falls to 57% among Labour voters. The discrepancy can be explained by a generational divide: while 75% of those over 65 – the demographic most likely to support the Tories – are judged certain to vote, that falls to 15% among 25- to 34-year-olds, and just 10% among 18- to 24-year-olds. These figures seem profoundly questionable – they would herald a truly catastrophic collapse in turnout – but they underline that Labour’s fortunes depend on who is motivated to vote. In the latest poll conducted by YouGov – a pollster which offers some of Labour’s worst results – the Tory gap fell from 17 to 12 points in four days. The shift was partly explained by Labour supporters – particularly at the younger end – being more likely to vote.

Some Labour MPs fret about the young vote. “The student vote is not as energised as it was in 2017,” says one; another says that among their local students there is “still pretty good support, but the enthusiasm for us has dissipated”. But it was only after the 2017 manifesto launch that the campaign truly caught fire last time. It is notable too that 1.35 million under-35-year-olds have registered to vote during the first two weeks of this campaign, almost twice the rate of 2017.

As it was last time, Labour’s manifesto is its one major opportunity to overcome the formidable odds stacked against it. For the party to have a chance of evicting Boris Johnson from No 10, that manifesto must put rocket boosters under its efforts to convince the under-40s to trudge out of their homes in mid-December to polling stations. It must address student debt, the housing crisis, low pay, the lack of secure skilled work, childcare, rip-off mediocre public transport, cuts to education budgets and the climate emergency. It must also tell a story of how those younger people who can’t rely on the bank of mum and dad have been locked out of their country’s future; and of how the resources and talent exist to satisfy their ambitions and provide a prosperous future for their children, but that the wealth they help create ends up in the pockets and offshore accounts of the rich.

Back in 2017, younger people saw Corbyn as an unlikely champion of a generation whose living standards and values had been systematically assaulted. In the remaining three weeks, the Labour leader has to show more fight to again convince them he is their tribune. That doesn’t mean Labour should abandon older generations: even convincing a small sliver of pensioners will help tip the balance. But there is an unquestionable political truth: that if insufficient young people turn up to vote, Johnson will triumph and our country will be in the throes of hard Brexit by the end of January. The party has little time, and so much to do.

• Owen Jones is a Guardian columnist