Wright's Friedman conjecture for random hyperbolic surfaces

Let $X$ be a random compact hyperbolic surface of genus $g$, sampled according to the Weil–Petersson probability measure. Write $N_X^\Delta(0,a)$ for the number of eigenvalues of the Laplacian $\Delta_X$ in the interval $(0,a)$, counted with multiplicity. Wright's conjecture. For any sufficiently small $\varepsilon>0$,

$$\lim_{g\to+\infty}\mathbb{P}^{\mathrm{WP}}_g\left(N_X^\Delta\left(0,\frac14-\varepsilon\right)=1\right)=1.$$

Equivalently, for every fixed sufficiently small $\varepsilon>0$, with high probability the only eigenvalue below $\frac14-\varepsilon$ is the trivial eigenvalue $0$. This is the random-hyperbolic-surface analogue of Friedman's theorem for random regular graphs; the conjecture concerns the spectral gap under Weil–Petersson measure and remains open in the source.