You could argue that Independent readers do not vote Tory, so are not worried about the leadership fight. With a politically sectarian press in this country, it always amuses me when a newspaper offers "helpful" advice to a party it campaigns to keep out of power.
The Mail, though, is nothing if not Conservative. It made the mistake of going for quick gratification in the early stage of the leadership election, by persuading "big beast" Ken Clarke to announce his candidacy by way of a Mail exclusive in exchange for some editorial support.
The Mail does not like losers and was embarrassed at backing one. It is sulking, and putting its energies into finding a sniff-and-tell story about the young pretender.
The Tory press, as we used to call it, has not come well out of the contest so far. The papers have shown themselves out of tune with Conservative MPs, unable to spot the emergence of young David (or youngish Liam) and unable to detect the desire for a new beginning among Tory supporters. It remains to be seen if these papers are as out of tune with the party membership in the country who will decide which David gets the prize.
Most interesting of all has been the performance of the Murdoch papers, and the entry into the leadership contest of "mystery" American Irwin Stelzer, not as a candidate but as an "influence". Where might you expect to read such words as these? "Nor is it good enough to promise tax cuts. Which taxes are to be cut? If the name of the game is growth, it will have to be those taxes that discourage people from working harder and taking the gamble of starting businesses. That will probably mean cutting taxes on high earners."
Quite apart from the language and subject matter, these sentiments are not those usually served to Sun readers, who are rarely the highest earners and the most entrepreneurially inclined.
Yet on page two on Thursday, the day of the second ballot, under the headline "Big ideas are not enough", The Sun publishes "Commentary by Irwin Stelzer, leading US economist". I doubt many eyes were diverted left from page three, but why was this piece there, rather than in, say, The Times where it would have looked less out of place?
When I worked for The Sunday Times, looking after the comment section of the paper, I would occasionally receive a call from the editor, Andrew Neil, telling me to expect a piece from Stelzer. I learnt that this man was closer to Rupert Murdoch than anyone on earth, that he was rich beyond compare, that he lived in crooner Andy Williams's former mansion in Aspen, Colorado. Stelzer is director of economic policy studies at the Hudson Institute in Washington.
More importantly, and less formally, he has the ear of Murdoch, who regularly seeks his advice. Stelzer is as laissez-faire a free marketeer and right-wing economist as you could find.
He is also very charming and understated, as he would demonstrate as he dropped by my desk from time to time.
Some Sun readers may be unaware of this background, may indeed spend little of their time in economic policy studies.
But there was the Stelzer piece in The Sun striking a cautious note about Cameron. Stelzer questioned whether Cameron had substance, whether he had thought through his economic policy. Cameron had a lot of work to do to persuade voters that he could go from a big idea (growth) to a workable programme.
The Sun has backed New Labour in elections since 1997, and its readers overwhelmingly vote Labour. Yet here it carries an unlikely piece that is essentially code for "Murdoch has his doubts about Cameron, and particularly his economic policy".
Cross refer to The Times on Friday. Its leader after the Cameron second ballot victory also struck a cautionary note. Cameron had enjoyed a meteoric rise "but meteors can fall to earth at speed as well". Cameron had withstood the heat of the supposed "drugs affair" with dignity, "but this is not the only aspect of his life and times of which the Conservative Party knows little".
And then there was the question of policy. "It would be dangerous for the Tories to assume that all they need to compete with Labour is a little stardust and hair oil. There also have to be new ideas."
The Times suggests that those should be articulating the virtues of competition and the market. Good to see The Times and The Sun singing from the same (Stelzer) hymn sheet.
Peter Cole is professor of journalism at the University of Sheffield.