Conjecture on the spectral distribution of random Schreier graphs

Let $q$ be a prime number, let $S$ be a random subset of $\textit{GL}_k(\mathbb{Z}_q)$, and let $G_k$ be the Schreier graph of $S \subset \textit{GL}_k(\mathbb{F}_q)$ acting on $(\mathbb{F}_q^k)^*$. Let $G'_k$ be a $2|S|$-regular graph from the permutation model of size $q^k-1$. Spectral distribution conjecture. As $k$ grows, the second largest eigenvalue distribution of $G_k$ converges to that of $G'_k$. The conjecture proposes that, in high dimension, random Schreier graphs have asymptotic second-eigenvalue statistics matching those of the permutation model, complementing the paper's experimental observations.