Wilde's quasi-simple commutator conjecture

Let $H$ be a finite group such that $L=[H,H]$ is quasi-simple and $\mathbf Z(L)=\mathbf Z(H)$, and let $\chi\in\operatorname{Irr}(H)$ be faithful. Wilde's quasi-simple commutator conjecture. For every $h\in H$ with $\chi(h)\ne 0$ and $H=\langle L,h\rangle$,

$$o(h\mathbf Z(H))\mid |H:\mathbf Z(H)|/\chi(1).$$

This is presented as an equivalent formulation of the nearly simple reduction and is the condition studied in the paper for fixed primes and various families of simple groups. It remains open in the exceptional cases not covered by the paper's results.