Mixed involution and odd-prime count conjecture

Let $S$ be a non-abelian simple group, let $G$ be a group, and let $I_r(X)$ denote the number of elements of order $r$ in a group $X$. Let $p$ be an odd prime divisor of $|S|$.

Mixed counting conjecture. If

$$I_2(G)=I_2(S)\quad\text{and}\quad I_p(G)=I_p(S),$$

then

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

The conjecture is presented as a consequence suggested by Herzog's conjecture and the cited conjecture on the values $I_p(G)$. No resolution is supplied in the paper excerpt.