Uniform finite-exceptional-spectrum conjecture for dense generators of compact Lie groups

Let $G$ be a simple compact Lie group, let $k\geq 4$, and let $S$ be a symmetric set of size $k$ generating a dense subgroup of $G$. Let $\Delta^S$ denote the associated averaging or adjacency operator. Uniform finite-exceptional-spectrum conjecture. There exists $\epsilon>0$ such that $\Delta^S$ has only finitely many eigenvalues outside the interval

$$[-k+\epsilon,k-\epsilon].$$

This is proposed as a stronger spectral-gap conjecture for dense generating sets; the preceding discussion relates spectral gaps to weak Diophantine properties, while uniformity over all such sets remains open.