The HRT conjecture for Schwartz functions

Let $S(\mathbb{R})$ denote the Schwartz space. For a nonzero $g\in S(\mathbb{R})$ and a finite set $\Lambda=\{(a_k,b_k)\}_{k=1}^N\subset\mathbb{R}^2$, define

$$\mathcal{G}(g,\Lambda)=\{e^{2\pi i b_k\cdot}g(\cdot-a_k)\}_{k=1}^N.$$

Schwartz-function sub-conjecture. For every nonzero $g\in S(\mathbb{R})$ and every such $\Lambda$, the system $\mathcal{G}(g,\Lambda)$ is linearly independent in $L^2(\mathbb{R})$.

This is a stronger regularity-restricted case of the HRT conjecture. The paper states that it remains open even for Schwartz functions, despite results for special point configurations and other special classes of functions.