Tame approximation conjecture for Grunwald realizations

Let $G$ be a finite group and let $S$ be a finite set of places of $\mathbf{Q}$, none dividing $|G|$. For each $v\in S$, let $F_v/\mathbf{Q}_v$ be a Galois extension equipped with an embedding

$$\operatorname{Gal}(F_v/\mathbf{Q}_v)\hookrightarrow G.$$

Tame approximation conjecture. There exists a Galois extension $L/\mathbf{Q}$ such that

$$\operatorname{Gal}(L/\mathbf{Q})\cong G$$

and

$$L_v\cong F_v^{\oplus\deg(L)/\deg(F_v)}$$

for every $v\in S$. This is a tame form of the Grunwald problem: local extensions should be simultaneously realized by a global extension with the prescribed Galois group. The supplied text describes related positive results for supersolvable groups and notes unresolved difficulties in general, so the conjecture remains open.