Prime-order generation conjecture for finite nonabelian simple groups

Let $G$ be a finite nonabelian simple group. Define

$$\operatorname{oe}(G)=\{m\mid \exists\,g\in G\text{ with }|g|=m\}$$

to be the set of element orders of $G$. Prime-order generation conjecture. There exist two distinct primes $p,q\in\operatorname{oe}(G)$ such that, for all $x,y\in G$ with $|x|=p$ and $|y|=q$, the subgroup $\langle x,y\rangle$ is nonsolvable; perhaps, even, it is nonabelian simple. The preceding theorem establishes the analogous assertion with $p$ and $q$ replaced by arbitrary element orders, while the conjecture asks whether the orders can always be chosen to be distinct primes. It is motivated by the verified cases for alternating and sporadic groups; its status is not established in the supplied text.