Spectral convergence conjecture for random covers of negatively curved surfaces

Let $X$ be a closed negatively curved surface with universal cover $\widetilde{X}$, and let $\lambda_0=\lambda_0(\widetilde{X})$ be the bottom of the $L^2$-spectrum of the Laplacian on $\widetilde{X}$. Let $X_n$ be the random covers of $X$ considered in the paper, and write $\mathrm{Sp}(\Delta_{X_n})$ and $\mathrm{Sp}(\Delta_X)$ for their Laplace spectra, counted with multiplicity. Spectral convergence conjecture. For every $\varepsilon>0$, with high probability as $n\rightarrow +\infty$,

$$\mathrm{Sp}(\Delta_{X_n})\cap \left[0,\lambda_0-\varepsilon\right]=\mathrm{Sp}(\Delta_X)\cap \left[0,\lambda_0-\varepsilon\right],$$

where the multiplicities coincide on both sides. This conjecture predicts that, with high probability, random covers introduce no new spectrum below the bottom of the universal-cover spectrum, up to an arbitrary margin. The paper proves the analogous statement with the smaller interval ending at $\lambda_0/2-\varepsilon$; the conjectured extension to $\lambda_0-\varepsilon$ remains open.