The Débes–Deschamps conjecture on finite split embedding problems

Let $k$ be a Hilbertian field. A finite split embedding problem over $k$ is an epimorphism

$$\alpha:G\rightarrow \operatorname{Gal}(L/k),$$

where $G$ is finite, $L/k$ is Galois, and there is an embedding $\alpha':\operatorname{Gal}(L/k)\rightarrow G$ with $\alpha\circ\alpha'=\operatorname{id}_{\operatorname{Gal}(L/k)}$. A solution is an isomorphism $\beta:\operatorname{Gal}(F/k)\rightarrow G$ for a Galois extension $F/k$ containing $L$, such that $\alpha\circ\beta$ is the restriction map to $\operatorname{Gal}(L/k)$. Débes–Deschamps conjecture. Each finite split embedding problem over any Hilbertian field has a solution. The conjecture would imply the inverse Galois problem over every Hilbertian field and the Shafarevich conjecture for the maximal cyclotomic extension of $\mathbb{Q}$. It is known in some important cases, including when the base field is a rational function field in one variable over an ample field, but is not established in general.