Ekström–Persson conjecture on dimensions of random limsup sets

Let $u$ be a probability measure on $\mathbb{R}^d$, let $(\omega_k)_{k=1}^{\infty}$ be independent random points distributed according to $\nu$, and for $\alpha>0$ define

$$E_{\alpha}(\omega):=\limsup_{k\to\infty}B(\omega_k,k^{-\alpha}).$$

Write $f_\nu(\alpha)$ for the almost sure value of $\operatorname{dim}_{\mathrm{H}}E_\alpha(\omega)$. If $F_\nu$ is the lower multifractal spectrum of $\nu$ and $\overline{F}_\nu$ is its increasing $1$-Lipschitz hull, then Ekström–Persson conjecture. For every $\alpha>0$,

$$f_\nu(\alpha)=\overline{F}_\nu\left(\frac{1}{\alpha}\right).$$

The conjecture predicts that the Hausdorff dimension of the random limsup set is determined by the lower multifractal spectrum of the sampling measure. It was stated for general probability measures on $\mathbb{R}^d$; the supplied source does not establish its resolution.