Will they call the whole thing off?
Certainly relations must have been strained over the past week. Rarely can a Prime Minister's attempt to restore his Government's fortunes have been so comprehensively scuppered as they were by the leak of the Education and Employment Secretary's briefing paper - a paper probably written for Mrs Shephard by her political adviser, Elizabeth Cottrell, and probably little different from the version she finally approved.
The timing of the leak was particularly damaging. Released to coincide with - and overshadow - last Thursday's Cabinet strategy meeting at Chequers, it followed only two days after the Prime Minister's speech to heads of grant-maintained schools, in which he set out incentives to boost the Government's opting-out drive.
Enter Mrs Shephard's paper, in which she makes clear her view that the real problem with the education service is lack of money. And the paper comments: "the need to improve standards must not be overshadowed by arguments about the mechanics through which education is delivered".
There is surely more to this than the usual attempt by a spending minister to bolster her bid for more money, although the outcome of the current round of public spending talks is especially crucial to the Education and Employment Secretary.
Last year, a letter leaked to The TES from somewhere in Whitehall contained her warning to Cabinet colleagues that up to 10,000 teachers' jobs could be lost if spending was not increased. She lost that argument and her warnings were to some extent borne out, although the worst impact of the cuts was cushioned by the advice to local authorities to spend their reserves.
But reserves, like political credibility, get used up. Since last year's defeat at the hands of the Chancellor, Mrs Shephard has been going around guaranteeing teachers and local authorities better luck next time, using half-promises by the Prime Minister to add weight to her words.
This time, she must deliver. Early reports suggest that her first encounter with William Waldegrave, Chief Secretary to the Treasury, was sticky. He offered an extra Pounds 400 million: she demanded an extra Pounds 800m. Perhaps the leak will help to bridge the gap.
But the paper makes a rather more fundamental point. It contains a clear message to the Prime Minister to stop placing such stress on political initiatives such as vouchers and opting out and concentrate, instead, on standards and resources in all schools.
It is here that the division lies. While Mrs Shephard is trying to retain her credibility with teachers, the Prime Minister has his eye on the next general election. Having promised peace and stability in education for five years, he now sees the need to regain the initiative on policy to show that the Tories have not run out of ideas.
The Education Secretary may reasonably feel she was given a false prospectus. When her old friend offered her the education portfolio last year, he made it clear her main job was to bring calm and reassurance to an education service rattled by constant change. None better than the competent, friendly, slightly schoolmistressy Mrs Shephard to do that. Mr Major gave the same message to the Conservative party conference last October.
Now, a year later, it seems he cannot have enough change. Perhaps the long skirmish between Number Ten and the Department for Education and Employment over nursery vouchers has deepened the Prime Minister's natural impatience with the educational establishment: it is said that he sent a number of the department's schemes back to the drawing board. He is certainly frustrated by the way the main Tory policy initiative - opting out - is losing momentum. And he is constantly having his radical resolve stiffened by a policy unit that sees itself as the guardian of Conservative policy against "the departmental view".
Norman Blackwell, head of the Policy Unit, is keenly interested in education, as is his new deputy, Dominic Morris, who has taken over the education brief. Mr Morris, although a career civil servant who has served at the DTI, moved straight from Mrs Thatcher's private office to a desk in the Policy Unit without passing again through Whitehall. "He's on our side," a right-wing MP said last week.
It was almost certainly Mr Morris who wrote the Prime Minister's speech to GM heads last week, possibly after consulting such insiders as Chris Woodhead, the Chief Inspector, who is said to visit the Number 10 Policy Unit.
To right-wingers and staff at the Policy Unit, Mrs Shephard is seen as having "gone native", putting the interests of her department and the education service before party policy, like innumerable education secretaries before her.
One right-wing insider claimed last week that, on education, Mr Major was "every bit as Thatcherite as Mrs Thatcher ever was", pointing out that the Prime Minister, unlike his predecessor, had never been Education Secretary and had therefore never encountered "the education world".
On this view, the PM remains strongly populist in his instincts, while the Education Secretary, as a former teacher and inspector, has drawn closer to a world to which she is naturally sympathetic. But it might be misleading to imagine that a deep policy rift has opened up between the two, neither of whom is, after all, a natural ideologue. It is more a question of emphasis and presentation.
One departmental insider suggested last week: "There is no great gulf between Gillian Shephard and John Major. But John Major is being egged on by a unit that is very pro-change and Gillian Shephard is being held back by a department that is very anti."
Nor is it sensible to suggest that Mrs Shephard herself had the document leaked out of a sense of fury at being thwarted and sidelined. If she had wanted to leak, she would have prepared the ground and timed it better. As it is, while the leak may help her in her battle for funds, it is of benefit only to the Labour party and lands her with the blame for doing further damage to the Conservatives' election prospects.