Uniform spectral gap conjecture for random covers of Anosov surfaces

Let $M$) be a closed Anosov surface, and let $M_n\to M$ be a uniformly random cover. Write $\gamma_0$ for the constant in the essential spectral-gap theorem and call a resonance of $M_n$ new if it does not arise from the base surface $M$.

Uniform spectral gap conjecture. For any $\varepsilon>0$, with probability tending to $1$ as $n\to +\infty$, the cover $M_n\to M$ has no new resonance in the half-space

$$\{z: \operatorname{Re}(z)>-\frac{\gamma_0}{2}+\varepsilon\}.$$

The conjecture would give a high-frequency localization result uniform over random covers, beyond the finiteness supplied by the essential spectral gap theorem. The source does not state a resolution.