Prison cells of ancient Athens: did Socrates die here?

Reader Sam Baxton very kindly sent me these pictures from his trip to Athens.  You're looking at the city's oldest known prison cells.  The question is, was Socrates in one of them?

The cells are embedded in Philopappou Hill.  That's the modern name.  You probably know it better as the Hill of Muses.

The cells are strongly believed to be carved out by hand.  Certainly the left and right ones are.  The middle one looks older.  In fact I used it as the template for the cell in which Nico finds himself incarcerated in The Pericles Commission.  Here's a closer look at it:

Needless to say, the Greek tourism authority has a sign outside calling this place Socrates' Prison.  The fact is, no one knows.  The cells might be a much later date.

Classical Athens had no need of a prison, because there was no such thing as a prison sentence.  A court could kill you, fine you, exile you, or let you go.  Those were the only options.  There was however a holding cell.

We know about the holding cell because Plato mentions it in his description of the death of Socrates.  He says the cell was within easy walking distance of the Acropolis.  These cells definitely qualify.  He also says Socrates was kept in the cell for an inordinate number of days because the Athenians very cleverly condemned him right before a sacred period when executions were forbidden.  He had to wait it out.

Then they "released him from his chains" on the day he was due to die.  That implies he might have walked out if it weren't for the chains.  That in turn makes it sound like the holding cell was some room at the agora, rather than any of these secure looking holes in a hill.

The counter-argument to the agora theory is that a simple room in the middle of Athens is probably not the best option if a condemned man has friends who are handy with a chisel.  Also you'd think someone, some time, might have described a cell in the city centre.

So no one knows!  But if Socrates was held in a special cell, then this is probably it.