The Vo-set two-invariant characterization conjecture for finite simple groups

Let $G$ be a finite group. Define

$$\operatorname{Van}(G)=\{g\in G\mid \text{there exists an irreducible complex character }\chi\text{ such that }\chi(g)=0\},$$

and let $\operatorname{Vo}(G)$ be the set of element orders of elements in $\operatorname{Van}(G)$. Let $S$ be a finite simple group.

Vo-set characterization conjecture. $G\cong S$ if and only if

$$\operatorname{Vo}(G)=\operatorname{Vo}(S)$$

and

$$|G|=|S|.$$

This was posed as an open problem in the Kourovka notebook; the supplied text gives no resolution status.