Rathmair's conjecture on local stability and Cheeger constants for the Gabor transform

Let $f\in L^2(\mathbb{R})$ be such that its Gabor transform $\mathcal{G}f$ is exponentially concentrated, meaning that there exists $\epsilon>0$ such that

$$|\mathcal{G}f|e^{\epsilon |z|}\in L^2(\mathbb{R}^2).$$

Rathmair's conjecture. The local stability and Cheeger constants of $f$ should be finite and non-zero.

This conjecture concerns the computational tractability of the constants governing local stability in Gabor phase retrieval. The surrounding discussion relates these constants to Cheeger inequalities for the metric conformally weighted by the magnitude of the Gabor transform; the supplied text does not state whether the conjecture has been resolved.