TITLE: Evidence for $Q^{\operatorname{solv}}$ being pseudo-algebraically-closed
QUESTION [12 upvotes]: 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?

REPLY [4 votes]: I'll respond to this question with another question.  Given a curve C/Q, there is a solvable extension K/Q such that C/K has local points everywhere.  (At least I think this is is the case; it requires some kind of argument on approximating local solvable extensions by global ones, which I've seen done before.) Some people believe that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for curves.  If that were the case, would it say anything about your question?