Near-optimal spectral-gap meta-conjecture for random hyperbolic surfaces

Let $(X_k)_k$ be a sequence of random finite-area hyperbolic surfaces such that

$$\lim_{k\to\infty}\operatorname{Vol}(X_k)=+\infty.$$

Write $\lambda_1(X_k)$ for the first positive Laplace eigenvalue. Random hyperbolic-surface spectral-gap meta-conjecture. For every $\epsilon>0$,

$$\lim_{k\to+\infty}\mathbb{P}\left(\lambda_1(X_k)\geq\frac14-\epsilon\right)=1.$$

The claim predicts that random hyperbolic surfaces asymptotically achieve the universal-cover threshold $1/4$ with high probability. It is presented as a meta-conjecture motivated by the analogy with random regular graphs and Selberg's conjecture; the source does not give a resolution.