Affine Fourier-spectrum threshold conjecture for distance sets

Let $d\geqslant 2$, let $\mu$ be a Borel probability measure on $\mathbb{R}^d$, and let $\dim_{\mathrm{F}}^\theta\mu$ denote its Fourier spectrum at parameter $\theta\in[0,1]$. Define

$$T_d^{\mathrm{conj}}(\theta)=1+\left(\frac d2-1\right)\theta, \qquad 0\leqslant\theta\leqslant 1.$$

Affine Fourier-spectrum threshold conjecture. If, for some $\theta\in[0,1]$,

$$\dim_{\mathrm{F}}^\theta\mu\geqslant T_d^{\mathrm{conj}}(\theta),$$

then

$$\dim_{\mathrm{H}}D(\operatorname{supp}\mu)=1.$$

This conjecture proposes the optimal affine threshold interpolating between the relevant endpoint values for Fourier spectrum and full-dimensional distance sets. The supplied text does not state whether it has been resolved, so its database status remains open.