Universal Gaussian Gabor frame-condition asymptotics and hexagonal optimality

Let $\Lambda(\alpha)\subset\mathbb{R}^2$ be a lattice of density $\alpha>1$, and let $\mathcal{G}(\varphi,\Lambda(\alpha))$ be the associated Gaussian Gabor system. Denote by $\kappa_\Lambda(\alpha)$ the condition number of its frame operator. Let $C_3$ be the constant appearing in the hexagonal-lattice asymptotics.

Universal Gaussian Gabor asymptotic conjecture. There exists a constant $C_\Lambda$, depending on the geometry of $\Lambda(1)$, such that

$$\kappa_\Lambda(\alpha)-\frac{C_\Lambda}{1-\frac{1}{\alpha}}\to0$$

as the critical density is approached from above, $\alpha\to1^+$. Moreover,

$$C_3\leq C_\Lambda,$$

with equality if and only if $\Lambda$ is the hexagonal lattice.

This proposes a universal near-critical asymptotic law for arbitrary lattices and identifies the hexagonal lattice as the unique optimizer. The source presents the general statement as an open problem motivated by known rectangular-lattice results and lattice-energy bounds.