Tracy–Widom limit and Ramanujan proportions for random regular graphs

Let $\mathcal{F}_{N,d}$ be one of the families of $d$-regular graphs $\mathcal{CI}_{N,d}$, $\mathcal{SCI}_{N,d}$, or $\mathcal{SCB}_{N,d}$, and let $\lambda_\pm(G)$ be the largest non-trivial positive and most negative eigenvalues. Write

$$\widetilde{\lambda_\pm}(G)=\frac{\lambda_\pm(G)-2\sqrt{d-1}+c_{\mu,N,d,\pm}N^{m_\pm(\mathcal{F}_{N,d})}}{c_{\sigma,N,d,\pm}N^{s_\pm(\mathcal{F}_{N,d})}}.$$

Tracy–Widom and proportion conjecture. The distribution of $\lambda_\pm(G)$, normalized in this way, converges as $N\to\infty$ to the $\beta=1$ Tracy–Widom distribution rather than a normalized $\beta=2$ or $\beta=4$ Tracy–Widom distribution or the standard normal distribution. For non-bipartite graphs, $\lambda_+(G)$ and $\lambda_-(G)$ are statistically independent. The constants satisfy $c_{\mu,N,d,\pm}<0$ and $s_\pm(\mathcal{F}_{N,d})>m_\pm(\mathcal{F}_{N,d})$, so approximately $52\%$ of graphs in bipartite families and $27\%$ otherwise are Ramanujan, meaning $\lambda(G)\leq 2\sqrt{d-1}$. The bipartite percentage is $51.9652\%$ to six digits, and the non-bipartite percentage is its square. This conjecture predicts the limiting edge-eigenvalue laws and hence the asymptotic proportion of Ramanujan graphs in the specified families; the source presents it as motivated by numerical evidence, and no proof or resolution is supplied.