Stronger frame-bound conjecture for optimal Gaussian Gabor lattices

Let $g_0(t)=2^{1/4}e^{-\pi t^2}$ be the Gaussian, let $\Lambda\subset\mathbb{R}^2$ be a lattice of fixed density $\delta$, and let $A$ and $B$ denote the sharp lower and upper frame bounds of $\mathcal{G}(g_0,\Lambda)$. For separable lattices write $\Lambda=\alpha\mathbb{Z}\times\beta\mathbb{Z}$, so that $(\alpha\beta)^{-1}=\delta$. Stronger frame-bound conjecture. For fixed density $\delta$, the lower frame bound of $\mathcal{G}(g_0,\Lambda)$ is uniquely maximized by the hexagonal lattice, and the upper frame bound is uniquely minimized there. For fixed density in the separable case, the lower frame bound is uniquely maximized and the upper frame bound is uniquely minimized if and only if

$$\alpha=\beta=\delta^{-1/2}.$$

This is stronger than the condition-number conjecture because it optimizes the two frame bounds separately and implies the claimed optimal conditioning. The source records that the separable assertion is proved for special densities, but does not establish the full conjecture.