Strohmer's conjecture on Gaussian Gabor frame-condition asymptotics

Let $\kappa_3(\alpha)$ and $\kappa_4(\alpha)$ denote the frame conditions of the Gaussian Gabor systems with hexagonal lattice $\Lambda_2(\alpha)=\alpha^{-1/2}H\mathbb{Z}^2$ and von Neumann lattice $\Lambda_{1\times1}(\alpha)=\alpha^{-1/2}\mathbb{Z}^2$, respectively, both of density $\alpha$. Let

$$C_r=\frac{2\pi r}{\tan(\frac{\pi}{r})}\frac{\Gamma(\frac{2}{r})^2}{\Gamma(\frac{1}{r})^4},\qquad r=3,4.$$

Strohmer's conjecture. As the critical density is approached from above, $\alpha\to1^+$,

$$\kappa_3(\alpha)-\frac{C_3}{1-\frac{1}{\alpha}}\to0 \qquad\text{and}\qquad \kappa_4(\alpha)-\frac{C_4}{1-\frac{1}{\alpha}}\to0.$$

These asymptotics were proved for the hexagonal and rectangular cases in the cited work; the conjectural part is the proposed exact constants. The source gives no resolution of the conjectural constants.