The Fock-space formulation of the HRT conjecture

Let $F^2$ be the Fock space, let $f\in F^2\setminus\{0\}$, and let $\lambda_1,\ldots,\lambda_N$ be distinct points in $\mathbb C$. For each $\lambda_k$, let $U_{\lambda_k}f$ denote the corresponding Fock-space translate.

HRT conjecture. The functions

$$U_{\lambda_k}f,\qquad 1\le k\le N,$$

are linearly independent in $F^2$. This is equivalent to the time-frequency formulation through the Bargmann transform. The conjecture remains open in general, although special cases are known and are sufficient for the affirmative results proved in the paper.