Characterization of non-abelian simple groups by singular-element proportions

Let $G$ be a finite group, and let $S$ be a non-abelian simple group. Write $\mu(G)$ for the set of proportions of $r$-singular elements of $G$ as $r$ ranges over the primes dividing $|G|$. Singular-element characterization conjecture. If

$$\mu(G)=\mu(S),$$

then $G\cong S$. The theorem established in the paper proves this characterization for $S={\operatorname{PSL}}(2,q)$ with $q\geq 4$; the conjecture asks whether the same rigidity holds for every non-abelian simple group.