Dominance of the continuous spectrum for random hyperbolic surfaces of large genus

Let $\mathbb{P}_{g,k(g)}$ be the probability measure on hyperbolic surfaces of genus $g$ with $k(g)$ cusps, and let $X \in \mathcal{M}_{g,k(g)}$ be a random surface. Assume that $k(g)=o(\sqrt{g})$ as $g\to\infty$, and let $I\subset(\frac14,+\infty)$ be a compact interval. Denote by $N(X,I)$ and $M(X,I)$ the discrete and continuous spectral contributions in the interval $I$, respectively. Dominance of the continuous spectrum. With high probability as $g\to\infty$,

$$\frac{N(X,I)}{M(X,I)}=o(1).$$

This would show that the continuous part of the spectrum is generically dominant for random hyperbolic surfaces of large genus, making the quantum ergodicity result primarily an equidistribution theorem for Eisenstein series. As far as the authors know, this problem is open.