Generalized Strohmer–Beaver conjecture for Gaussian Gabor frames

Let $g_0(t)=2^{1/4}e^{-\pi t^2}$ be the standard Gaussian, let $\delta>1$, and define

$$\mathfrak{F}_{\Lambda}^{\delta}(g_0)=\{\Lambda\subset\mathbb{R}^2\mid \Lambda\text{ is a lattice and }\operatorname{vol}(\Lambda)^{-1}=\delta\}.$$

For each $\Lambda$ in this class, let $A$ and $B$ denote the lower and upper frame bounds of the Gabor frame operator $S_{g_0,\Lambda}$. Generalized Strohmer–Beaver conjecture. Among all lattices in $\mathfrak{F}_{\Lambda}^{\delta}(g_0)$, the hexagonal lattice is the unique maximizer for the lower frame bound and the unique minimizer for the upper frame bound. This conjecture implies the Strohmer–Beaver condition-number conjecture, while the source records that the latter was disproved; the status of this stronger formulation is not established by the supplied evidence.