Separable Gaussian Gabor frame conjecture for the square lattice

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

$$\mathfrak{F}_{(\alpha,\beta)}^{\delta}(g_0)=\{\alpha\mathbb{Z}\times\beta\mathbb{Z}\mid \alpha,\beta\in\mathbb{R}_+,\ (\alpha\beta)^{-1}=\delta\}.$$

For each separable lattice in this class, let $A$ and $B$ denote the lower and upper frame bounds of its Gabor frame operator. Square-lattice conjecture. Among all separable lattices in $\mathfrak{F}_{(\alpha,\beta)}^{\delta}(g_0)$, the square lattice is the unique maximizer for the lower frame bound and the unique minimizer for the upper frame bound. The conjecture is the separable-case question that remains after the earlier square-lattice optimality conjecture was disproved in the unrestricted setting; the supplied evidence does not resolve this separable formulation.