Logarithmic biorthogonal-norm estimate for the Gaussian Gabor system

For $(m,n)\in\mathbb{Z}^2$, let $\gamma_{m,n,0,0}$ denote the biorthogonal element associated with the Gaussian Gabor atom indexed by $(m,n)$, and let $c_{m,n}>0$. The Gaussian Gabor system is formed from the atoms $e^{2\pi \int}e^{-\pi(t-m)^2}$. Logarithmic estimate conjecture. When $(a,b)=(0,0)$, one has

$$\left\|\gamma_{m,n,0,0}\right\|_{L^2(\mathbb{R})}^2\asymp\log\left(e+\sqrt{m^2+n^2}\right).$$

Consequently, if

$$\sum_{(m,n)\in\mathbb{Z}^2\setminus\{(0,0)\}}\frac{\log\left(e+\sqrt{m^2+n^2}\right)}{c_{m,n}^2}<\infty,$$

then the weighted Gabor system

$$\left\{c_{m,n}e^{2\pi \int}e^{-\pi(t-m)^2}:(m,n)\in\mathbb{Z}^2\setminus\{(0,0)\}\right\}$$

is a complete Riesz–Fischer sequence and a lower semi frame for $L^2(\mathbb{R})$. This is motivated by preliminary symbolic and numerical exploration, but the author explicitly has not verified the estimates, so the claim and its consequence remain open.