Conjectural essential spectral gap bound for twisted Selberg zeta functions

Let $X$ be a compact hyperbolic surface, let $\mathcal{H}^1(X)$ denote its space of harmonic $1$-forms, and let $G_\omega$ be the essential spectral gap associated with $\omega\in\mathcal{H}^1(X)$. Write $\operatorname{Pr}(\omega)$ for the pressure appearing in the definition of this gap, and let $Z_\omega$ and $\phi_j$ denote the associated twisted Selberg zeta function and eigenfunctions, respectively.

Essential spectral gap conjecture. For every $\omega\in\mathcal{H}^1(X)$,

$$G_\omega\geq\frac{1}{2}\bigl(\operatorname{Pr}(2\omega)-1\bigr).$$

In particular, $G_\omega=0$ if and only if $\omega=0$; hence every nonzero real harmonic $1$-form gives failure of the asymptotic version of the Riemann hypothesis for $Z_\omega$ and failure of quantum unique ergodicity for $\phi_j$. The passage presents this as a conjecture motivated by Jakobson–Naud and gives no resolution.