The relativized Alon conjecture for random graph covers

Let $B$ be a fixed graph, let $\widehat B$ be its universal cover, and let $A_{\widehat B}$ be the adjacency operator on $\widehat B$. Let $\mathcal{C}_n(B)$ denote the model of degree-$n$ covering graphs of $B$, and let $\rho^{\mathrm{new}}_B(A_G)$ be the new adjacency spectral radius of a cover $G$ relative to $B$. The relativized Alon conjecture. For every $\epsilon>0$,

$$\Pr_{G\in\mathcal{C}_n(B)}\left[\rho^{\mathrm{new}}_B(A_G)\ge \rho(A_{\widehat B})+\epsilon\right]\longrightarrow 0$$

as $n\to\infty$. This asserts that the new spectrum of a random high-degree cover is asymptotically bounded by the spectral radius of the universal cover. The source attributes the conjecture to earlier work and presents results toward it; no resolution is supplied here.