Built a skeleton piece of giant coax in EZNEC, been playing with it as
a monopole over MININEC ground.

I'm not going to claim much since it's probably wildly inapplicable,
but I will claim that the answer to this question ain't trivial and
depends on the details of the ends.

I find a current in the center conductor, It's very much bigger when
the center conductor is slightly extended past the ends of the "shield"
than when it's slightly inside. The shield is eight wires arranged in
a regular octogon with the tops and bottoms tied together. It shows
more or less normal fat monopole behavior when used alone. (33 foot
element resonant around 6.4 MHz)

A fatter wire used for the center conductor has greater current than a
thinner one. A very thin wire exhibits almost no current.

The peak current in the center conductor is about 3% of what's flowing
in the shield for a 1 foot "diameter" (the circumscribed circle around
the octagon) shield and a 6 inch diameter inner conductor. The current
is very low at either end of the center conductor and peaks about 2/3
from the top.

This dumb model is full of holes of many kinds, I'm sure, but it does
seem to show nontrivial effects of changing things a little bit near
the ends of the coax (slight extension of center conductor outside
of/retraction of center conductor into the shield)

Dan