Time-frequency shift invariance conjecture for Gabor spaces with an $S_0$-generator

Let $g\in S_0({\mathbb R})$ and let $\Lambda \subset {\mathbb R}^2$ be a lattice such that $(g,\Lambda)$ is a Riesz basis for its closed linear span $\mathcal G(g,\Lambda)$. Time-frequency shift invariance conjecture. The time-frequency shifts $T_a M_b$ that leave $\mathcal G(g,\Lambda)$ invariant have their parameters $(a,b)$ in $\Lambda$. This conjecture asks whether the rational-density result extends to arbitrary lattices; the cited rational-density theorem relies on the Zak transform, and the irrational-density case remains open.