Lubotzky's bounded-generator expander conjecture for finite simple groups

Let $(G_i)$ be the family of all non-abelian finite simple groups. A Cayley graph $\mathcal{C}(G_i,S_i)$ is the graph associated with the generating set $S_i$, and a family of graphs is an $\epsilon$-expander family when it has a uniform expansion constant $\epsilon>0$. Lubotzky's conjecture. There exist generating sets $S_i$ of uniformly bounded size such that the Cayley graphs

$$\mathcal{C}(G_i,S_i)$$

form a family of $\epsilon$-expanders for some fixed $\epsilon>0$. The theorem proved in the paper establishes this for alternating and symmetric groups. Further evidence comes from expander families of fixed Lie type and from logarithmic-diameter generating sets, but uniform expansion across all non-abelian finite simple groups remains open in the source.