  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.
