The two-orders characterization conjecture for finite simple groups

Let $G$ be a finite group and $S$ a finite simple group. Write $\pi_e(G)$ for the set of element orders of $G$.

Two-orders characterization conjecture. $G\cong S$ if and only if

$$\pi_e(G)=\pi_e(S)$$

and

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

Thus every finite simple group should be characterized by the order of the group and the orders of its elements. This conjecture was proposed by the author in 1987; the supplied text gives no resolution status.