Strong Larson's conjecture for functions tiling by translations and dilations

Let $f:\mathbb{R}\to\mathbb{R}$ be a nonnegative function that tiles $obreak\mathbb{R}$ by translations and dilations, meaning

$$\sum_{j\in\mathbb{Z}}f(2^jx)=1\quad\text{a.e.},\qquad \sum_{k\in\mathbb{Z}}f(x+k)=1\quad\text{a.e.}$$

A wavelet set is a measurable set $E\subset\mathbb{R}$ satisfying the corresponding translation and dilation tiling conditions. Strong Larson's conjecture. There exists a wavelet set contained in the support of $f$.

This strengthens Larson's conjecture from Fourier supports of wavelets to arbitrary functions satisfying both tiling identities. The source does not specify whether it is open or resolved.