Spectral approximation conjecture for Harper operators on amenable graphs

Let $X$ be a graph with a free action by an amenable group $amma]$, with finite fundamental domain. Let $\{X_m\}_{m=1}^{\infty}$ be the regular exhaustion of $X$ corresponding to a regular exhaustion $\Lambda_m$ of $\Gamma$. For each $m$, let $N_m$ be the size of $\Lambda_m$, and let $E_m(\lambda)$ denote the number of eigenvalues less than or equal to $\lambda$ of the DML restricted to $X_m$, with either Dirichlet or Neumann boundary conditions. Let $F$ be the spectral density function of the DML on $X$, defined using the von Neumann trace.

Spectral approximation conjecture. The normalized eigenvalue-counting functions converge to the spectral density function:

$$\lim_{m\rightarrow\infty}\frac{E_m(\lambda)}{N_m}=F(\lambda),\qquad\forall\lambda\in\mathbb{R}.$$

This conjecture asserts that finite-volume restrictions of the DML recover the spectral density of the infinite graph, independently of whether Dirichlet or Neumann boundary conditions are used. Its status is not determined by the supplied text.