Conjecture on Airy point-process fluctuations of higher eigenvalues

Let $G_n$ be the hexagonal-lattice rhombus and let $H_n$ be its weighted adjacency operator in either the i.i.d. or mixed model. Let $\bar f_\gamma$ and $\bar g_\gamma$ be the constants from the mixed-model theorem, and let $\lambda_{n,k}$ be the $k$th smallest positive eigenvalue of $H_n$. Define

$$\alpha_{n,k}=\frac{-\log \lambda_{n,k}-\bar f_\gamma n}{(n\bar g_\gamma/2)^{1/3}}.$$

Higher-eigenvalue Airy-process conjecture. For any fixed $k$, as $n\to\infty$, the tuple $(\alpha_{n,1},\ldots,\alpha_{n,k})$ converges in distribution to the top $k$ points of the Airy point process. This conjecture extends the proved Tracy–Widom fluctuation result for the smallest positive eigenvalue to any fixed number of higher positive eigenvalues in both models.