Sarnak's conjecture on Ramanujan random regular graphs

Let $G_{n,d}$ be a uniformly random simple $d$-regular graph, and write

$$\lambda(G)=\max\{|\lambda_2|,|\lambda_n|\}$$

for the maximum absolute value of its nontrivial adjacency eigenvalues. A $d$-regular graph is Ramanujan when $\lambda(G)\le 2\sqrt{d-1}$. Sarnak's conjecture. For fixed $d$ and $n{\tilde o}\infty$, $G_{n,d}$ is Ramanujan with positive constant probability. Random regular graphs are natural candidates for producing Ramanujan graphs, but the source gives no resolution of this probabilistic assertion.