Wilde's nearly simple group conjecture

Let $G$ be a finite group such that $S\lhd G/\mathbf Z(G)\leq\operatorname{Aut}(S)$ for some non-abelian simple group $S$, and let $\chi\in\operatorname{Irr}(G)$ be faithful. Suppose that

$$\chi\text{ is irreducible over }G^{(\infty)},$$

and that $\chi(g)\ne 0$ and $G=\langle G^{(\infty)},\mathbf Z(G),g\rangle$ for some $g\in G$. Wilde's nearly simple conjecture. Then

$$o(g\mathbf Z(G))\mid |G:\mathbf Z(G)|/\chi(1).$$

This is a reduction of the strong form of Wilde's conjecture to nearly simple groups. The paper verifies the corresponding condition for many families, but leaves exceptional cases open, particularly cases involving extensions of irreducible characters.