Moretó's element-order-count characterization conjecture for finite simple groups

Let $S$ be a finite simple group, and let $p$ be the largest prime divisor of $|S|$. For a group $H$, let $H(p)$ denote the set of elements of order $p$; equivalently, the number of elements of order $p$ is $|H(p)|$. Moretó's conjecture. If $G$ is a finite group satisfying

$$|G(p)|=|S(p)|\quad\text{and}\quad |G|=|S|,$$

then $G\cong S$. The conjecture is false in general: the source gives counterexamples with $S=A_8$, $G=L_3(4)$ and with $S=O_7(3)$, $G=S_6(3)$, although it holds for all sporadic simple groups and for alternating groups except $A_8$ and $A_{10}$.