Conder's (2,3)- or (2,5)-generation conjecture

For a finite group $G$, being $(2,r)$-generated means that $G$ is generated by an involution and an element of order $r$. Conder's conjecture. Every non-abelian finite simple group is $(2,r)$-generated for some $r\in\{3,5\}$, except for ${\rm U}_3(3)$, which is $(2,7)$-generated.

The conjecture seeks an absolute bound on the prime orders needed for generation; the source states that it remains open, with the exceptional group ${\rm U}_3(3)$ requiring order $7$.