Uniform spectral convergence for operators on abstract quasicrystal graphs

Let $G(V,E)$ be a connected infinite graph with bounded vertex degrees. Assume that $G$ is amenable and is an abstract quasicrystal graph: for every radius $r$ and every $r$-pattern $\alpha$ in the finite set $P_r(G)$ of rooted-ball isomorphism classes, there is a frequency $P(\alpha)$ such that, for every Følner sequence $\{Q_n\}_{n=1}^{\infty}$, the proportion of vertices in $Q_n$ having pattern $\alpha$ converges to $P(\alpha)$. Uniform spectral convergence conjecture. For random Schrödinger operators, percolation Laplacians, and self-adjoint pattern-invariant operators associated to $G$, uniform spectral convergence exists. The claim concerns a common spectral-convergence phenomenon for several operator classes on graphs with uniform local-pattern frequencies; the source does not specify the precise meaning of uniform spectral convergence or provide evidence resolving the conjecture.