Shafarevich's conjecture on Galois groups over fields ramified at finitely many places

Question

Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$, $\overline{K}$ a fixed separable closure of $K$, and $G_K:=\mathrm{Gal}(\overline{K}/K)$ the absolute Galois group of $K$. Let $S$ be a finite set of non-Archimedean places of $K$, or equivalently, a finite set of non-zero prime ideals in the ring of integers $\mathcal{O}_K$ of $K$. We define $$\overline{K}_S:=\text{the maximal algebraic extension of $K$ in $\overline{K}$ unramified outside $S$}$$ and $$G_{K,S}:=\mathrm{Gal}(\overline{K}_S/K).$$

At the beginning of Barry Mazur's part of Modular Forms and Fermat’s Last Theorem, he states:

What is the "structure" of $G_{K,S}$ - whatever that means? It is not even known whether or not $G_{K,S}$ is finitely generated as a topological group (although this has been conjectured to be the case by Shafarevich about thirty years ago).

Does anyone have a source for this conjecture? Since this book was published in 1997, I guess the conjecture that Mazur is referring to would've been formulated in the mid-60s.

The only conjectures by Shafarevich that I could find are (1) the Tate-Shafarevich conjecture on the Tate-Shafarevich group, and (2) concerning the absolute Galois group of $\mathbb{Q}^{\text{ab}}$. I don't think either is equivalent to the above, but if so, what's the intuition behind that?

Answer 1

In the footnote on Page 11 of [Fernando Q. Gouvea: Deformations of Galois Representations],

The reference is

[I.R. Shafarevich, Algebraic number fields, Proceedings of the international Congress of Mathematicians, Stockholm, 1962 (Djursholm), Inst. Mittag-Leffler, 1963, Translated version reprinted in I.R. Shafarevich, Collected Mathematical Papers (Springer- Verlag, 1989), pp. 283-294, pp. 163-176.],

but we should note that there Shafarevich simply asks whether it is the case that $G_{K,S}$ is finitely generated for any number field (and whether the number of generators can be bounded in terms of the number of elements of $S$). His main reason for posting the question is that the analogous statement is true for function field over $\mathbb{C}$.