Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In Theorem 10.2.5 of Cohomology of Number Fields, it's proved that the group $ G_{K,S} $ is topologically generated by the conjugacy classes of finitely many elements, and hence $ G_{K,S} $ contains a proper dense normal subgroup if $ G_{K,S} $ is infinite. Note that if $ G_{K,S} $ is topologically finitely generated, then $G_{K,S}$ contains a countable dense subgroup.
Question: Is it true that the profinite group $ G_{K,S} $ contains a countable dense normal subgroup, at least at the level of conjecture?
I know that it's still an open problem whether $ G_{K,S} $ is topologically finitely generated. Any comments and reference would be highly appreciated.
