Labour MPs don’t know how to solve their electoral problem, but they know where it resides: in small English cities and towns, and anywhere voters with degrees are thin on the ground. In the 2017 general election, the party lost Mansfield, Stoke-on-Trent South, and Walsall North, and failed to win Bolton West, North Swindon, and Calder Valley in Yorkshire.

Politicians and sympathetic left-leaning think tanks are looking at the problem. On the Labour Party’s right, it has long been a preoccupation of Yvette Cooper and Rachel Reeves, among others; meanwhile Lisa Nandy, from the soft left, is the patron of a new think tank called the Centre for Towns. Most Labour politicians tend to identify the same solutions: a tougher stance on welfare and immigration, and a radical rebalancing of the British economy. (Reeves recently wrote a serious pamphlet on the latter, The Everyday Economy.)

The difficulty, however, is that Labour is already pursuing these policies. The party under the leadership of Jeremy Corbyn is committed to accepting many of the Conservatives’ welfare cuts and to ending the free movement of people, which would allow immigration to be reduced dramatically. And one thing you cannot say fairly of Corbyn – who has proposed the nationalisation of large parts of industry – is that he doesn’t have a radical plan to reshape the British economy.

Any objective analysis would discover that Labour’s true stumbling-block in regaining seats such as Mansfield is the antipathy that many small-town voters have for Labour in general and for the present leader in particular. But Corbyn’s hegemony inside the party machine means that Reeves and the others know that there is no point mentioning this fact.

Labour’s small-town problem is one of the few sources of Conservative solace amid the mess over Brexit and the ongoing psychodrama over Theresa May’s leadership. In the local elections on 3 May, Labour suffered net losses in the West Midlands and north-west England; the Tories believe that Corbyn cannot appeal to what one cabinet minister dubs the “British rust belt”.

Yet the reality is that Tory bullishness rests on shaky foundations. Corbyn improved his standing with voters at the last election, when they saw more of him on television, and he might yet repeat the trick at the next. And Brexit might trigger an economic and political crisis from which Labour is the beneficiary, regardless of any doubts voters have about its leader.

Although Corbyn would need to improve the geographic reach of his appeal to form a majority Labour government, there is another path to Downing Street, through a confidence and supply arrangement or even a minority administration. The former chancellor George Osborne used to say that Ed Miliband’s task in 2015 was the political equivalent of “water flowing uphill”.

The Conservatives, seeking a fourth consecutive term in office with a mouldy economy and a crumbling country, now face a similar challenge. In 2017 they were aided by a perfect storm in Northern Ireland, in which the DUP achieved its best ever result in terms of seats won (with ten), and Sinn Féin – which does not take its seats at Westminster and therefore makes the parliamentary arithmetic easier for minority governments – won a further seven.

At the next election, if either party is defeated in Northern Ireland by the non-sectarian Alliance or Labour’s sister party, the SDLP, then the Conservatives’ prospects for staying in office begin to look very poor indeed. And that’s before you factor in the potential for even a modest Liberal Democrat revival, or Labour learning how to use its huge and energised activist base to leverage narrow wins in the seats it lost by a small number of votes in 2017, such as Amber Rudd’s Hastings (majority: 346) or Justine Greening’s Remain-supporting Putney constituency.

In truth, Conservative confidence doesn’t rest on electoral maths but on an instinct or feeling: a sense in the pit of the Tory gut that voters will recoil from Corbyn in an election when he could realistically become prime minister. (Last time, few believed he could win.) Tories had a similar feeling about Miliband in 2015 and they were right. Their faith was rewarded with an unexpected majority for David Cameron, and this increases the Conservative belief that the same thing will happen next time.

In the Labour leader’s office, officials know full well that the party’s present position in the polls is disappointing at this point in the electoral cycle – given the poor economic data and the Tory chaos – and that the local election results were, at best, a partial success.

But they have faith, too, remembering how their impressive 40 per cent vote share on 8 June last year was preceded by securing a meagre 27 per cent in local elections the month before. Team Corbyn believes that popular Labour policies, such as rail renationalisation and action for renters – as well as their own dominance of the party machine – will allow them to perform even better next time, when it arrives.

What both parties are gambling on is that past performance is a guide to future success. Conservative MPs look at a forbidding path to re-election and a Labour leader they regard as inadequate and believe that the 2015 election will play itself out a second time, in their favour. Corbyn’s aides look at electoral challenges smaller than the ones they faced in 2017, and believe that a better result can be achieved. Meanwhile, Corbynsceptic Labour MPs fear that the true barrier to getting what they want doesn’t really reside in small towns in the West Midlands or north-west England – but in the office of the Labour leader whom they despise, but accept cannot be removed.