The center conductor isn't entirely floating.

If the shield were closed at both ends, there would be no fields in the
coax, but the coax center conductor at each end can have current
induced on it from the outside world.

I think you'd go about describing the coupling as being via "fringing
fields" at the ends if you were to think of a free end of coax with
fields in it radiating into space, I'm thinking of the reciprocal
behavior...

But this is a behavior where I'm imagining a differential-mode current
existing to start with...

I think the answer is highly influenced by the exact details of the end
of the coax and where it is with respect to other objects... the
coupling is very light and mostly to the shield at each end... and I
think this makes it not entirely unlike just putting the shield wire
and the center conductor wire in parallel in space with the ends tied
together.

I think current flows in the center conductor in phase with the current
in the shield and there's very little differential mode current if any.

Certainly I'm wrong, I'm going to think on it some more and try to
figure out how wrong...

What current flows in the center conductor if you short it to the
shield with one wire at each end? What about if you do it with a solid
metal cap ?

73,
Dan