Bilu–Linial's 2-lift conjecture

Let $G$ be a finite regular graph, and let $H$ be a 2-lift of $G$. An eigenvalue of $H$ not inherited from $G$ is called new, and let $\rho(\mathsf{UCT}(G))$ denote the relevant universal-cover spectral radius. Bilu–Linial's conjecture. For every regular finite $G$, there is a 2-lift $H$ of $G$ such that every new eigenvalue satisfies

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

The source notes that the stronger bound $|\lambda|\leq\rho(\mathsf{UCT}(G))$ was also conjectured, and that the stated conjecture would yield a tower of infinitely many such lifts. The supplied status evidence resolves the family of conjectures in this section by Marcus, Spielman, and Srivastava's interlacing-polynomial method.