Quartic-residue subgraph minimum-eigenvalue conjecture

Let $p$ be a prime for which the quartic residues are defined, let $F_p$ be the induced subgraph of the Paley graph $G_p$ on the nonzero fourth powers modulo $p$, and let $A_{F_p}$ be its adjacency matrix. Quartic-residue eigenvalue conjecture.

$$\lim_{p\to\infty}\frac{\lambda_{\min}(A_{F_p})}{\sqrt p}=-\frac12.$$

The quartic-residue subgraph has the same number of vertices as a degree-2 localization but exhibits different spectral behavior. The source explicitly reports a counterexample to the corresponding localized-graph prediction, while this more specific asymptotic is presented as plausible and remains unproved.