The HRT conjecture on linear independence of finite Gabor systems

Let $\Lambda\subset\mathbb{R}^{2}$ be a fixed finite set, let $f\in L^{2}(\mathbb{R})$ be nonzero, and write

$$\mathcal{G}(f,\Lambda)=\{M_yT_xf:(x,y)\in\Lambda\}.$$

Here $\mathcal{G}(f,\Lambda)$ is the associated finite Gabor system. HRT conjecture. The finite Gabor system $\mathcal{G}(f,\Lambda)$ is linearly independent. This conjecture concerns the basic question of whether nonzero time-frequency shifts can satisfy a finite linear relation. Despite substantial partial progress, the general assertion remains open.