Minimum-eigenvalue conjecture for localized Paley graphs

Let $G_p$ be the Paley graph of prime order $p$, let $a\geq1$, and let $\mathcal I_a(G_p)$ denote the independent sets of size $a$. For $I\in\mathcal I_a(G_p)$, let $A_{G_{p,I}}$ be the adjacency matrix of the localized graph and let $\lambda_{\min}$ denote its smallest eigenvalue. Minimum-eigenvalue conjecture.

$$\lim_{p\to\infty}\min_{I\in\mathcal I_a(G_p)}\frac{2^{a+1}}{\sqrt p}\lambda_{\min}(A_{G_{p,I}})=\lim_{p\to\infty}\max_{I\in\mathcal I_a(G_p)}\frac{2^{a+1}}{\sqrt p}\lambda_{\min}(A_{G_{p,I}})=-2\sqrt{2^a-1}.$$

This asserts uniform asymptotic spectral-edge behavior across all localizations. The paper proves the degree-$a=1$ case and presents the general statement as conjectural, with the maximum-eigenvalue analogue noted as a related possibility.