Babai–Kantor–Lubotzky conjecture on expanders from finite simple groups

Let $G$ be a nonabelian finite simple group. A subset $S\subset G$ is symmetric when it is closed under taking inverses, and $\operatorname{Cay}(G,S)$ denotes the Cayley graph of $G$ with generating set $S$. Babai–Kantor–Lubotzky conjecture. There are constants $k\in\mathbb{N}$ and $\lambda<1$ such that, for every nonabelian finite simple group $G$, there is a symmetric set $S\subset G$ of $2k$ generators for which

$$\operatorname{Cay}(G,S)$$

is a $\lambda$-spectral expander graph. This conjecture significantly guided research on expander graphs; its status is not specified in the source.