Support conjecture for tight-frame spectra

Let $mu$ be a compactly supported probability measure in ${\mathbb R}^d$, let $\widetilde{\mu}(E)=\mu(-E)$, and let $\Lambda$ be a tight-frame spectrum for $mu$. Write $\delta_\Lambda$ for the counting measure on $\Lambda$, and let the Fourier transform be understood in the tempered-distribution sense.

Support conjecture for tight-frame spectra. Then

$$\operatorname{supp}\widehat{\delta_\Lambda}\subset \{0\}\cup\bigl(\operatorname{supp}(\mu*\widetilde{\mu})\bigr)^C.$$

The conjecture arises formally by taking Fourier transforms in the tiling equation for $|\widehat{\mu}|^2$. If true, it would constrain the Fourier support of every tight-frame spectrum through the support of the self-convolution measure; the source does not establish it.