Heil's four-point HRT conjecture for a specific configuration

Let

$$\Lambda=\{(0,0),(0,1),(1,0),(\sqrt{2},\sqrt{2})\}\subset\mathbb{R}^2,$$

and let $g\in L^2(\mathbb{R})$ be nonzero with $\|g\|_2=1$. Define the associated Gabor system by

$$\mathcal{G}(g,\Lambda)=\{e^{2\pi i b\cdot}g(\cdot-a):(a,b)\in\Lambda\}.$$

Heil's four-point conjecture. The system $\mathcal{G}(g,\Lambda)$ is linearly independent.

This is a particular unresolved instance of the HRT conjecture for four points. The paper recalls it as a conjecture attributed to Heil; its status is not resolved in the supplied text.