Fix a finite set $S$ of places of $\mathbb Q$. Let $G_{\mathbb Q,S}$ be the Galois group of the maximal extension of $\mathbb Q$ unramified outside S$. I believe that it is an open question whether this group is topologically finitely generated, that is, contains a dense, finitely generated subgroup.

Is there a standard conjecture about whether $G_{\mathbb Q,S}$ should be finitely generated or not?

Has anyone published an opinion?

JSE mentions a stronger conjecture: $|S|$ generators suffice; further, the existence of a special set of generators labeled by $S$ and inert at the labeling place. Is this refinement standard? (Note that a finite extension would suffice to disprove this hypothesis.)

Weakening the hypothesis, we could restrict the ramification at $S$ to be tame.