Containment conjecture for spectra of universal covers and limiting eigenvalue supports

Let $J_0$ be a Jacobi matrix on a finite connected graph $\mathcal G_0$. Let $\mathcal{G}_n$ be a sequence of finite connected covers of $\mathcal G_0$ such that $|V(\mathcal{G}_n)| \to \infty$, and let $J_n$ be the lift of $J_0$ to $\mathcal{G}_n$. Let $J_T$ be the lift of $J$ to the universal cover $T$. Let $\nu_n$ be the normalized eigenvalue counting measure of $J_n$ and assume that the weak limit $\nu_\infty=\lim_{n \to \infty}\nu_n$ exists.

Containment conjecture.

$$\sigma(J_T)\subseteq \operatorname{supp}\nu_\infty.$$

This conjecture asserts that, for connected finite covers, the spectrum of the lifted Jacobi matrix on the universal cover is contained in the support of every limiting eigenvalue counting measure. It proposes that disconnected covers are the only obstruction to such containment; the statement remains open in the supplied source.