Linear independence conjecture for finite wavelet systems generated by Schwartz functions

Let $\phi\in\mathcal{S}(\mathbb{R})\setminus\{0\}$, and let a finite wavelet system generated by $\phi$ be a finite collection of wavelet dilates and translates of $\phi$. Finite-wavelet-system conjecture. Each finite wavelet system generated by a nonzero Schwartz function is linearly independent. The paper presents this as a conjecture following cases in which linear independence is proved, including three-point systems; the general assertion remains open.