NPE characterization conjecture for finite simple groups

Let $G$ be a group and $S$ be a finite simple group. Let $pe(H)$ denote the invariant used in the source, namely the combined data of the relevant element-order counts. NPE characterization conjecture. Then $G\cong S$ if and only if

$$\npe(G)=\npe(S)\quad\text{and}\quad |G|=|S|.$$

The source presents this as a strengthening of Moretó's conditions by incorporating the number of involutions. It does not provide a resolution of this conjecture in the supplied text.