Weak Larson conjectures on measurable subsets of supports

Let $\psi$ be a dyadic orthonormal wavelet, let $f:\mathbb{R}\to\mathbb{R}$ tile $obreak\mathbb{R}$ by translations and dilations, and let $\mu$ denote Lebesgue measure. Define a measure $\nu$ on Lebesgue-measurable sets by

$$\nu(E)=\int_E\frac{1}{|x|}\,d\mu(x).$$

A weak Larson conjecture consists of the following two assertions: (1) the support of $\psi$ contains a measurable set $V$ with $\mu(V)=1$ and $\nu(V)=\log 4$; and (2) the support of $f$ contains a measurable set $V$ with $\mu(V)=1$ and $\nu(V)=\log 4$.

The source presents these assertions as consequences of Larson's conjecture and notes that a wavelet set has the stated measure values. Their resolution is not specified.