Here by triangle group $(a,b,c)$ I mean the group with presentation $$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$
In other words, for every finite simple nonabelian group $G$, do there exist pairwise coprime integers $a,b,c$ such that $G$ is generated by $x,y$ with $|x| = a$, $|y| = b$, and $|xy| = c$?
I'd also be interested in any result where we relax the "pairwise-coprime" condition to the condition that $(|x|,|y|)\cdot (|x|,|xy|)\cdot (|y|,|xy|)$ is small.