Bouc's nilpotence conjecture for the beta invariant of finite groups

Let $G$ be a finite group. The invariant $\beta(G)$ is defined in Bouc's theory of $B$-groups.

Bouc's conjecture. $\beta(G)$ is nilpotent if and only if $G$ is nilpotent.

This conjecture was proposed by Bouc and concerns the relationship between the nilpotence of a finite group and that of its associated invariant $\beta(G)$. The supplied text does not state whether the conjecture has been resolved.