Jakobson–Naud's essential spectral gap conjecture

Let $X=\Gamma\backslash\mathbb{H}$ be a convex co-compact hyperbolic surface, let $\delta$ be the dimension of the limit set of $\Gamma$, and define the essential spectral gap by

$$G(X):=\inf\Bigl\{\sigma<\delta:\ \mathcal{R}_X\cap\{\operatorname{Re}s\geq\sigma\}\text{ is finite}\Bigr\}.$$

Jakobson–Naud's essential spectral gap conjecture.

$$G(X)=\frac{\delta}{2}.$$

Jakobson and Naud formulated this notion and proved the lower bound $G(X)\geq\frac{\delta(1-2\delta)}{2}$ for convex co-compact groups. The asserted equality remains open in the source.