Infinitely many $D$-Ramanujan graphs under the second-eigenvalue condition

Let $D$ be a degree matrix of order $t\geq 2$, let $T_D$ be its universal cover, and let $\lambda_2(D)$ denote the second eigenvalue of $D$. A finite graph with degree matrix $D$ is $D$-Ramanujan when its $(k+1)$-st eigenvalue is at most $\rho(T_D)$, where $k$ is the largest integer such that $\lambda_k(D)\geq\rho(T_D)$. Second-eigenvalue infinitude conjecture. If

$$\lambda_2(D)<\rho(T_D),$$

then there exist infinitely many $D$-Ramanujan graphs. This is a more specific proposed abundance statement for degree matrices satisfying the displayed spectral separation; the source gives no resolution.