Are the Conservatives panicking? Not quite yet, perhaps, but a little tense? They should be. Most of the polls are showing Labour closing a gap once thought impossibly wide. An election that was theirs to lose is slipping away from them, and if they can hold on to their rapidly eroding lead until June 8 they will be very relieved.

Is a Labour victory likely? Still not. But it's no longer unthinkable, and some of the reasons why were on display last night when they were each interviewed by a studio audience and by Jeremy Paxman for Sky News and Channel 4.

They covered familiar ground. On Brexit, May said she would walk away from divorce talks with the European Union without a deal if she had to, but Corbyn said he would make sure an agreement was reached if he won power. May struggled to respond to questions about cuts, and was heckled by the audience. Corbyn was pressed on corporation tax and security.

The Prime Minister did not shine, though she didn't make a complete hash of it; she is no Diane Abbott. She was rather wooden and spoke in slogans. Jeremy Corbyn, on the other hand, was articulate and engaging, and his answers had real substance. Of the two, there's no doubt who was the winner – not only in the audience debate, but in the subsequent trial by Paxman. The once-feared interviewer asked Corbyn a series of foolish blind-alley questions the Labour leader handled well enough; May was clearly rattled when he challenged her on whether the EU would see her as a 'blowhard who collapses at the first sign of gunfire' after she softened her plans on old age care.

On the other hand, neither is there any doubt about which has the higher mountain to climb. Two years ago, when Corbyn was first elected, I wrote a piece that I think is worth quoting (it was headed 'Jeremy Corbyn for Labour leader: Right ideas, completely wrong choice').

'A Corbyn-led government would not see the streets running red with the blood of plutocrats. His economic plan involves raising taxes on rich people and businesses, reducing corporate tax relief and using the money to support infrastructure projects. He also wants to crack down on tax avoidance (don't we all?) and reverse cuts to HMRC staff to help them collect more taxes.

'...However, Corbyn is also associated with various other causes that are polling-booth poison. He wants dialogue between Britain and Argentina on the Falklands. He wants to re-nationalise the railways. He supports the Palestine Solidarity Campaign. He supports a united Ireland and invited Gerry Adams and other Sinn Fein members to Parliament in 1984 just after the Brighton bombing. He's referred to Hamas and Hezbollah as "friends". In short he's given away so many hostages to fortune that he can barely field an army. Oh, and he's also anti-Trident, anti-military intervention and generally pro-peace.

'There's nothing indefensible about any of these positions, individually. But you can see Labour's problem. Faced with a Tory machine that is the political equivalent of a Great White shark, it's going to need a much, much bigger boat.'

Well: the Great White shark has looked more like a goldfish during the last couple of weeks, but apart from that the caveats still stand. Corbyn is most vulnerable on his record, and the right-wing press has remorselessly retailed stories of his unfortunate meetings and pronouncements over the years. It's these – some picked up by Paxman in his inquisition – that have led to his huge unpopularity among the voting public, who tend to be older. His warmth and personality, on the other hand, with his passionate denunciations of austerity, appeal more to the largely non-voting younger generation. It's not a good trade.

The truth is that last night both Corbyn and May did what they do. Of the two, Corbyn is most likely to benefit because he was the better performer by far. Whether it was enough to sway the electorate sufficiently remains uncertain – and brain fades like the one on Women's Hour this morning, when he forgot how much Labour's childcare plans would cost, won't help. But there is a week of campaigning to go, and there's everything to play for.

Follow Mark Woods on Twitter: @RevMarkWoods