Wilde's conjecture on zeros of characters and orders of elements

Let $G$ be a finite group and let $\chi\in\operatorname{Irr}(G)$. Here $o(g)$ denotes the order of $g$, and $|G|/\chi(1)$ is the codegree of $\chi$. Wilde's conjecture. If $\chi(g)\ne 0$ for some $g\in G$, then

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

This conjecture connects character zeros and element orders, and implies the theorem of Brauer and Nesbitt. The paper reduces it to statements about nearly simple groups and establishes strong forms in many cases, but the remaining cases require currently unavailable information on extensions of irreducible characters.