Elisabeth Murdoch has as much chance of landing the BBC top job as Katie Price becoming Archbishop of York

In the steeplechase for the director-generalship of the BBC, not one but two of the horses appear to be Caligula’s.

Whenever a viscerally startling personnel choice is mooted or made, dullards like me ritually trot out the late emperor’s equine bestie, Incitatus.

The reference is so mind-tranquilisingly overused that it operates as a sort of mythic ketamine on the anxious reader. But for anyone somehow unfamiliar with it, legend holds that the emperor promoted the beast to the high Roman office of consul.

Whether he did or did not (see below), the tale has been the yardstick by which every spectacularly demented public appointment has been gauged for almost two millennia.

Of late, Donald Trump has invoked it frequently (John Bolton as national security adviser, the Mooch’s marathon stint at comms, etc). Boris Johnson’s picks for his first cabinet franked the Baby Trump form (Priti Patel as home secretary, and so on).

One could go on and on, but what would be the point? If you gathered together every straitjacket-qualifying hire ever made, or floated, you’d struggle to find anything to equal the prospect of Elisabeth Murdoch becoming BBC director-general.

Correction: in suggesting earlier that the idea of Rupert Murdoch’s daughter leading the corporation her sire has devoted 40 years to annihilating was uniquely ridiculous, I was mistaken.

In fact, a second rumour renders the notion tediously conventional, and to that bombshell we’ll come below.

But first to Liz, whose qualifications for the grandest and most challenging post in UK media seem to rest with having started (and later sold to Daddy’s News Corp for the doubtless rigorously exact market value of $673m) the TV production company Shine.

A glance at Shine’s output is instructive. Who can forget the revival of Gladiators, sadly brief as it was, for Rupert’s Sky One? Or a dance reality show, cunningly entitled Got To Dance, for the same network?

As for The Biggest Loser, the ITV weight loss extravaganza hosted by both Kate Garraway and Davina McCall, sound TV analysts identify that meisterwork as the civilization of its day.

This sparkling body of work apart, the one factor adduced in her favour is that she, unlike her old man, has publicly supported the licence fee now under grave threat.

In her defence, relations with Rupert are thought to be strained in other areas. Small wonder if so. Liz had a beloved horse herself as a small girl, she told a magazine, until she came home from school to find it absent from the paddock of the family’s home counties farm.

Distraught, she rushed inside and asked Rupert what was going on? Not to worry, darling, he reassured her, but he’d given it away as first prize in a News of the World readers’ competition.

I’m no follower of Freud (Sigmund, that is; not her ex-husband Matthew), but that kind of lark can damage a father-daughter bond. So can a long, attritional battle with rival siblings for the succession if and when he departs the planet.

But however un-Rupert-like she might be, Liz Murdoch surely has as realistic a chance of this job as Katie Price has of being offered the Archbishopric of York.

While the source of this fantasy isn’t known, a hunch suspects it emanated from Downing Street to terrorise the BBC Trust (packed with government appointees) towards picking someone to the government’s authoritarian tastes with the threat of someone worse.

Speaking of which brings us to that yet more surreal rumour. Also cited in an almost entirely female field, and from the same stable, is a certain Rebekah Brooks.

She too is known as a lover of horses, though more as procuress than jockey after that curious business with the retired Metropolitan Police nag she borrowed on David Cameron’s behalf back in the halcyon time when, with Liz and Matthew, the pair headed the Cotswold glitterati.

I needn’t labour the incalculable absurdity of this one. Suffice to say, that next to Brooks – the surrogate daughter he gave that £14m pay off when she was charged over phone-hacking, and re-employed after she left the dock without a stain on her character – Rupert’s biological one looks oddly like Lady Reith.

Tremulously, in the unnerving knowledge that nothing nowadays can confidently be ruled out, let it be stated that neither of these fabled frenemies has the remotest chance of being offered the job, or would touch it if they were.

But then Caligula never appointed his horse as consul. He floated the possibility, reliable historians of the period posit, as a satirical expression of his contempt for the senate, knowing that it would rubber stamp whoever, or whatever, he proposed.

The cravenness of the BBC, which has never recovered from the trauma of the Hutton report, has been plain from its news and current affairs output for a while. It desperately needs a leader with the balls to resist the Johnson-Cummings axis bullying, rebuild its atrophied reputation for fearless impartiality, and protect the licence fee.

The next director-general almost certainly will be neither the genetic or commercial spawn of Rupert’s fecund loins. But the fact that the corporation hasn’t quashed such wildly preposterous rumours suggest that whoever it appoints will be anything but a paradigm of courage.