Uniform expansion conjecture for quasirandom groups

Let $G$ be a finite group. Say that $G$ is $\varepsilon$-quasirandom if every non-trivial irreducible complex representation of $G$ has dimension at least $\varepsilon^{-1}$; a Cayley graph of $G$ is connected when its generating set generates $G$, and it is an $\varepsilon$-uniform expander** when its expansion constant is bounded below uniformly in the generating set by a quantity depending on $\varepsilon$ only.

Uniform expansion conjecture. If $G$ is $\varepsilon$-quasirandom, then all connected Cayley graphs of $G$ are $\varepsilon$-uniform expanders.

This extends Breuillard's conjecture from finite simple groups of Lie type of bounded rank to all quasirandom groups and all generating sets. The source states that the analogous conjecture for bounded-cardinality generating sets is equivalent, via Theorem~, to Breuillard's conjecture; the general conjecture remains open.