Larson's conjecture on wavelet sets in Fourier supports

Let $obreak\thetap$ be a dyadic orthonormal wavelet, and let $u$ be its Fourier transform. A wavelet set is a measurable set $E\subset\mathbb{R}$ such that

$$\bigcup_{k\in\mathbb{Z}}(E+k)=\mathbb{R}\quad\text{a.e.},\qquad \bigcup_{j\in\mathbb{Z}}2^jE=\mathbb{R}\quad\text{a.e.}$$

Larson's conjecture. The support of the Fourier transform of a dyadic, orthonormal wavelet contains a wavelet set.

The conjecture asks whether every dyadic orthonormal wavelet has a Fourier support containing a measurable set that tiles $obreak\mathbb{R}$ by integer translations and dyadic dilations. Its status is not specified in the source.