Generalized ten Martini conjecture for magnetic Schrödinger operators on Fuchsian groups

Let $\Gamma$ be a cocompact Fuchsian group and let $\sigma$ be a multiplier on $\Gamma$. Let $\theta$ denote the flux associated with $\sigma$, and let $H_{\sigma,V}$ be the corresponding magnetic Schrödinger operator for a potential $V\in\mathbb C(\Gamma,\sigma)$. Generalized ten Martini conjecture. If the flux $\theta$ is irrational, then there exists a potential $V\in\mathbb C(\Gamma,\sigma)$ such that $H_{\sigma,V}$ has infinitely many gaps in its spectrum. The conjecture concerns the still-mysterious spectral-gap structure of magnetic operators at irrational flux; analogous gap-labeling results are known in several settings, but the asserted existence of infinitely many gaps here is presented as an open problem.