Uniqueness of the extremal periodic configuration in the discrete random displacement model

Let $d\geq 2$, let $q\neq 0$ be sign-definite, and let $\omega^{\ast}$ denote the periodic configuration associated with the extremal spectral values $E_{\min}$ and $E_{\max}$. A periodic configuration is understood up to translation when it is identified with any of its lattice translates. Uniqueness conjecture. The configuration $\omega^{\ast}$ is, up to translation, the unique periodic configuration satisfying

$$\min \sigma(h_{\omega^{\ast}})=E_{\min},\qquad \max \sigma(h_{\omega^{\ast}})=E_{\max}.$$

This is the discrete analogue of a uniqueness result known for the continuum random displacement model. It remains open here because the discrete Schrödinger equation lacks the unique continuation properties used in the continuum argument.