Uniform essential spectral gap conjecture for congruence subgroups

Let $\mathcal{R}_q$ be the resonance set associated with the congruence subgroup $\Gamma(q)$, and let $\delta$ denote the Hausdorff dimension of the limit set. Uniform essential spectral gap conjecture. For every $\sigma>\delta/2$, the set

$$\mathcal{R}_q\cap\\{s:\operatorname{Re}(s)\geq\sigma\\}$$

is finite and independent of $q$. This conjecture proposes a uniform version of the Jakobson–Naud essential spectral gap property for the family of congruence subgroups. Existing uniform spectral-gap results give partial information, but the asserted finiteness and independence of $q$ remain open in the stated generality.