The quarter-gap conjecture for random hyperbolic surfaces with cusps

Let $\mathcal{M}_{g,n(g)}$ be the moduli space of hyperbolic surfaces of genus $g$ with $n(g)$ punctures, equipped with the Weil–Petersson probability measure, and let $\operatorname{spec}(X)$ denote the spectrum of a surface $X$. Assume

$$\lim\limits_{g\to\infty}\frac{n(g)}{\sqrt g}=0.$$

Quarter-gap conjecture. For every $\epsilon>0$,

$$\lim\limits_{g\to\infty}\operatorname{Prob_{WP}^{g}}\left(X\in\mathcal{M}_{g,n(g)};\ \operatorname{spec}(X)\cap\left(0,\frac{1}{4}-\epsilon\right)=\emptyset\right)=1.$$

This predicts that the lower bound in the established theorem for the regime $n(g)=o(\sqrt g)$ can be improved to the Selberg value $\frac14$, analogously to the conjectural optimal spectral gap for closed hyperbolic surfaces. The statement remains open.