Britain is on the cusp of being ruled by an old Etonian. Again. (Boris Johnson would be the school’s 20th former pupil to become prime minister.)

But this grimly inevitable prospect has pushed a group of Labour members into action. The Labour Against Private Schools campaign, which launched this week, has devised a policy for the main party to adopt. Its aim? For the party’s next election manifesto to pledge to integrate all private schools into the state sector – ie to phase out private schools entirely.

For about six weeks, three teachers – who teach in inner London, Manchester and Durham and are also Labour members – have built a national campaign and policy plan to abolish private schools. Their aim is to pass a motion in September at Labour’s annual autumn conference, where members vote on party policy, and change the party's current stance on private schools.

The policies in the motion include: ensuring universities admit the same proportion of private school students as in the wider population (7 per cent), nationalising the big, historic endowments of the major public schools, and withdrawing charitable status (and the accompanying tax advantages) from private schools.

The campaign has attracted support from Labour frontbenchers Clive Lewis and Laura Pidcock, and backbench MPs including Ed Miliband, Kate Green, Lloyd Russell-Moyle, Rachel Reeves, Thelma Walker and Laura Smith (the latter two are former teachers).

“The Tory leadership election has definitely focused our minds on the issue somewhat, given we have basically got this slightly strange Eton versus Charterhouse varsity match,” says Holly Rigby, who has taught English in inner London for six years and has long been a critic of private schools. She joined Labour when Jeremy Corbyn first stood for the leadership in 2015.

“There’s never really been any kind of serious campaign to take the issue on either inside or outside the Labour Party,” she says. “It feels like there is momentum behind people realising now that it is a bit rank, basically.”

Indeed, the old boys at the top of politics are mirrored throughout the UK establishment. Report after report from the social mobility group, the Sutton Trust, shows that the privately educated disproportionately dominate British public life, and the best-paid and most powerful jobs.

Two-fifths of the UK’s elite group, as defined in the latest Sutton Trust report Elitist Britain 2019 in June, were privately educated – more than five times the general population – while a quarter had graduated from Oxford or Cambridge.

It’s the kind of inequality you’d think Jeremy Corbyn’s Labour Party – so focused on “the many, not the few” – would be leading the charge against. And indeed, they talk a good game against private schools.

Shadow education secretary Angela Rayner said earlier this year of Eton’s charitable status: “Even as state schools suffer the worst cuts in a generation, an elite few still benefit from tax giveaways with the government’s blessing.” Rayner is instinctively opposed to the bought privilege that private education represents.

Her recent proposal to replace the Social Mobility Commission with a Social Justice Commission is a signal that Labour is relinquishing the polite myth that we live in a meritocracy (“social mobility”, the argument goes, is merely a leg-up for a lucky or talented few).

“For too long our top professions have been a closed club, dominated by a wealthy and privileged elite who attended the same private schools,” Rayner said, when the Sutton Trust report was published in June. “The old boys’ network and the old school tie still hold back talented and hard-working people from less privileged backgrounds.”

However, Labour’s only current policy offer in this area is to pay for universal free school meals for primary school children by introducing VAT on private school fees. That’s not to say there won’t be more to come. Although Labour has been quiet on reforming private schools beyond this policy, Rayner has previously met with MPs backing the current campaign, and the party’s frontbench are examining wider policies to advance social justice.

There is an appetite to go further, and a feeling among Labour policymakers that a Boris Johnson premiership could push the issue of private school privilege up the political agenda. This may lead the Labour's education team to intervene on the subject before Labour Against Private Schools’ motions even reach the conference floor.

“In the next general election Labour needs to really claw back its anti-establishment and insurgent message,” says Rigby. “And I think railing against private schools is actually a really good way to do that, because I think people in the country know it’s unfair. So I think it’s a no-brainer.”

But one of the main obstacles is money. While raising taxes from private schools by removing or restricting their charitable status is attractive to Labour in terms of revenue, abolishing the independent sector is less so for the same reason. There’d be nothing to tax, and the state would have to pay to educate many more children.

“There are a lot of priorities in education, and do we want to prioritise spending a lot of money on richer children?” asks Jim Knight, a peer and former Labour MP who was schools minister in 2006-09. “I don’t think we do, I think we want to spend it on poorer children.”

Phasing private schools out may chime with the current leadership’s values (Corbyn is known for his strident commitment to comprehensive education), but its policymakers would find it tough, particularly in view of shadow chancellor John McDonnell's commitment to cost all spending policies. It would mean abandoning that free school meals pledge, for one thing, as zero private schools means zero potential VAT on the fees.

The expense is partly why Labour has historically struggled to limit private education. Anthony Crosland – who had railed against the indifference of socialists towards the “glaring injustice of the independent schools” in the Fifties – failed to abolish private schools when he had the chance as education secretary under Harold Wilson. When later asked why, Crosland said that finding maintained places for the 6 per cent of children hitherto educated privately would have been a strange use of resources, considering the poor condition of state schools at the time.

A further barrier is legality. Preventing people from buying private education, or removing schools' charitable status (for example, from technically private schools that provide specialist teaching for children with certain disabilities), could be legally challenged. Labour policymakers don’t want to formulate plans that would likely be overturned in the courts.

“I thought about this in office,” says Knight. “We had the discussions – was there more we should do to get the private sector to earn their charitable status? You’d have to bring a law so that you changed education as a charitable purpose, and you’ve got to know very clearly where to draw the line. It becomes really challenge-able, because people will argue, legitimately, and ultimately in the courts, that you put it in the wrong place… There are deep pockets who can fund expensive lawyers on behalf [of some schools].”

The greatest challenge over the years, however, has been political. The most radical Labour proposal for private schools appeared in Michael Foot’s 1983 “longest suicide note in history” manifesto – and that was just a promise to end private schools' charitable status and to “integrate” them into the state sector “where necessary”, stopping short of abolition.

“We did weigh up, as the Charity Commission subsequently did, withdrawing VAT exemption – but in the end this was vetoed on political grounds,” says David Blunkett, former education secretary under Tony Blair.

Private school reform was consigned to the “too difficult drawer” when Labour was in government, reveals Knight.

“I think the politics at the time were regarded as too hard, and it’s very reasonable to argue that we were too scared of the Daily Mail ten years ago, I’m very happy to take that on the chin,” he tells me. “But I do think we did the right thing in the private school/state school partnerships, opening up facilities to state schools, gently encouraging the Charity Commission... But to go the whole hog and say essentially we’re going to nationalise them all; it’s really mad politics.”

What are those politics? “I think there’s a problem with the politics of envy, there’s a problem in the politics of your priorities,” says Knight. “Should you spend that money on fixing the problem of a few schools, or should you devote your energy to improving all schools for all children?”

Labour’s current leadership are less afraid of the politics, but the legal and financial obstacles of old remain.

“Only at this moment to use a Second World War analogy – to Napoleon as well – we could ‘open a Russian front’, which would win no friends and end up in failure – surely a definition of futility if ever I heard one,” says Blunkett.

Yet Rigby and her campaign group are optimistic that the free school meals proposal – which she calls “an amazing policy” – could pave the way for more radical reform.

“We really hope now that Labour would go much further, because that [the current policy] would barely make a dent really, as would removing charity tax status,” she says. “We support that – but it wouldn’t make a dent on the big, elite public schools.”