On the non-frame property of Gabor systems with Hermite generators and the frame set conjecture

The frame set conjecture for Hermite functions formulated in [Gr\"ochenig, J. Fourier Anal. Appl., 20(4):865-895, 2014] states that the Gabor frame set for these generators is the largest possible, that is, the time-frequency shifts of the Hermite functions associated with sampling rates $\alpha$ and modulation rates $\beta$ that avoid all known obstructions lead to Gabor frames for $L^{2}(\mathbb{R})$. By results in [Seip and Wallst\'en, J. Reine Angew. Math., 429:107-113, 1992] and [Lemvig, Monatsh. Math., 182(4):899-912, 2017], it is known that the conjecture is true for the Gaussian, the $0$th order Hermite functions, and false for Hermite functions of order $2,3,6,7,10,11,\dots$, respectively. In this paper we disprove the remaining cases except for the $1$st order Hermite function.