Mohar's quasi-Ramanujan conjecture for multipartite trees

Let $D=(d_{ij})$ be a degree matrix for a multipartite graph, and let $\mathbb{T}_D$ be the unique infinite tree with degree matrix $D$. An infinite graph is weakly Ramanujan if it has at least one finite quotient that is $X$-Ramanujan, and it is $k$-quasi-Ramanujan if infinitely many finite quotients have their $(k+1)$st eigenvalue at most the spectral radius of the universal cover. For a degree matrix $D$, define

$$k_D=\max\{k:\lambda_k(D)\geq\rho(\mathbb{T}_D)\}.$$

Mohar's conjecture. For every degree matrix $D$, if $\mathbb{T}_D$ is weakly Ramanujan, then it is $k_D$-quasi-Ramanujan; moreover, if $k_D=1$, then $\mathbb{T}_D$ is Ramanujan. These conjectures concern the existence of infinitely many finite quotients with optimal spectral behavior for multipartite universal covers. The supplied status evidence says that several conjectures from this section, including these Mohar conjectures, were proven by Marcus, Spielman, and Srivastava using interlacing polynomials.