This is a follow-up to the following answer:

Solvable class field theory

in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed (this means, in particular, any geometrically connected variety over Q has a point over a solvable extension).

I am curious what evidence there is to support such a conjecture.

In addition, what can be said for the analogous statement for global function fields?

This is a follow-up to the following answer:

Solvable class field theory

in which it is stated as a "folklore" conjecture that the maximal solvable extension of Q is pseudo algebraically closed (this means, in particular, any geometrically connected variety over Q has a point over a solvable extension).

I am curious what evidence there is to support such a conjecture.

In addition, what can be said for the analogous statement for global function fields?