The Schwartz-function version of the HRT conjecture

Let $(t_j,\xi_j)_{j=1}^n$ be $n\ge 2$ distinct points in the plane, and let $f:\mathbb R\to\mathbb C$ be a Schwartz function. A nontrivial linear dependence means that not all coefficients $d_i$ vanish and

$$\sum_{j=1}^n d_i f(x+t_j)e^{2\pi i\xi_jx}=0$$

for almost every $x\in\mathbb R$. Schwartz-function HRT conjecture. There is no nontrivial Schwartz function satisfying such a nontrivial linear dependence. This is a weaker conjecture than the $L^2$ version and is described in the source as having been circulated in the literature; the source does not state a resolution.