The Lubotzky–Weiss generating-set conjecture for expander Cayley graphs

Let $\Gamma_i$ be a family of finite groups, and let $\Sigma_i$ and $\Sigma_i'$ be generating sets of $\Gamma_i$ whose sizes are uniformly bounded. Lubotzky–Weiss generating-set conjecture. If the Cayley graphs $\mathcal{C}(\Gamma_i,\Sigma_i)$ form an expander family, then the Cayley graphs $\mathcal{C}(\Gamma_i,\Sigma_i')$ should also form an expander family. This conjecture was disproved by Alon, Lubotzky, and Wigderson using zig-zag products, so the property of being an expander can depend on the generating set.