Strohmer and Beaver's conjecture on optimal Gaussian Gabor lattice geometry

Let $g_0(t)=2^{1/4}e^{-\pi t^2}$ be the Gaussian, and let $\Lambda\subset\mathbb{R}^2$ be a lattice of fixed density $\delta(\Lambda)=\delta>1$. For a Gaussian Gabor system $\mathcal{G}(g_0,\Lambda)$, write $A$ and $B$ for its sharp lower and upper frame bounds. A lattice is hexagonal when it has the form

$$\Lambda_h=\delta^{-1/2}QM_h\mathbb{Z}^2,$$

where $Q$ is orthogonal and

$$M_h=\sqrt{\tfrac{2}{\sqrt{3}}}\begin{pmatrix}1&\frac{1}{2}\\0&\frac{\sqrt{3}}{2}\end{pmatrix}.$$

Strohmer and Beaver's conjecture. Among Gaussian Gabor systems with fixed lattice density greater than $1$, the condition number $\operatorname{cond}(S_{g_0,\Lambda})=B/A$ is minimal if and only if $\Lambda$ is a hexagonal lattice. In the special case of rectangular lattices $\Lambda_{(\alpha,\beta)}=\alpha\mathbb{Z}\times\beta\mathbb{Z}$ with $(\alpha\beta)^{-1}>1$ fixed, the square lattice, characterized by $\alpha=\beta$, minimizes the condition number.

This conjecture proposes hexagonal geometry as optimal for the conditioning of Gaussian Gabor frames, with the square lattice as the optimal restriction among separable lattices. The source notes that the separable assertion was proved for special densities in 2017, while the full statement is not thereby resolved.