Babai's polylogarithmic diameter conjecture for finite simple groups

Let $G$ be a nonabelian finite simple group, and let $\operatorname{diam}(G)$ denote the maximum of $\operatorname{diam}(G,X)$ over all generating sets $X$ of $G$. Babai’s conjecture.

$$\operatorname{diam}(G)\ll(\log|G|)^{O(1)}.$$

This is the principal conjectural diameter bound for finite simple groups. The source attributes it to Babai, while noting that it first appeared jointly with Seress. The conjecture remains open for $A_n$ and for other simple groups of unbounded rank.