Schacher's Sylow metacyclic conjecture for rational admissibility

Let $G$ be a finite group. A group is Sylow metacyclic if all of its Sylow subgroups are metacyclic. Schacher's conjecture. The group $G$ is $Q$-admissible if and only if $G$ is Sylow metacyclic.

This conjecture proposes a complete description of the finite groups admitting a crossed-product division algebra over $\mathbb{Q}$. The source gives no resolution status.