2 problems

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

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 $L^2(\mathbb{R})$.