Clark's infinite-lift conjecture for finite graphs

Let $G$ be a finite graph, and let $H$ be a lift of $G$. Write $\mathsf{spec}(G)$ for the spectrum of $G$, and let $\rho(\mathsf{UCT}(G))$ denote the relevant universal-cover spectral radius. An eigenvalue of $H$ in $\mathsf{spec}(H)\setminus\mathsf{spec}(G)$ is called new. Clark's conjecture. For every finite $G$, there are infinitely many lifts $H$ of $G$ such that every new eigenvalue $\lambda\in\mathsf{spec}(H)\setminus\mathsf{spec}(G)$ satisfies

$$\lambda\leq\rho(\mathsf{UCT}(G)).$$

This generalizes the Bilu–Linial conjecture from 2-lifts to arbitrary lifts and would produce infinitely many Ramanujan graphs in relevant universal-cover settings. The supplied source does not state a resolution of this conjecture; its status is therefore left open.