The orthonormal wavelet implies wavelet-set conjecture

Let $A$ be an invertible matrix, let $B$ be its transpose, and let $\Gamma$ be a full-rank lattice in $\mathbb{R}^n$ with dual lattice $\Gamma^*$. An orthonormal wavelet for $(B,\Gamma^*)$ is an orthonormal wavelet associated with this dilation and translation pair, and an $(A,\Gamma)$ wavelet set is a measurable set whose integer $A$-dilates and $\Gamma$-translates each tile $\mathbb{R}^n$ measurably.

Orthonormal wavelet implies wavelet-set conjecture. For each pair $(B,\Gamma^*)$ such that there exists a $(B,\Gamma^*)$ orthonormal wavelet, there exists an $(A,\Gamma)$ wavelet set, where $B$ is the transpose of $A$ and $\Gamma^*$ is the dual lattice of $\Gamma$.

This conjecture asks whether every pair admitting an orthonormal wavelet also admits a wavelet set. The paper notes that this implication holds in all cases currently known, while the general question remains open.