Thévenaz's solvability conjecture for finite groups and their largest B-group quotients

Let $G$ be a finite group, and let $\beta(G)$ denote its largest quotient that is a $B$-group. Thévenaz's conjecture. The group $\beta(G)$ is solvable if and only if $G$ is solvable. The source attributes this conjecture to Thévenaz through Bouc and does not report a proof or disproof, so its general status remains open.