Typical-measure multifractal spectrum conjecture for compact subsets of Euclidean space

Let $K\subset\mathbb{R}^{d}$ be a compact set, and let $\mathcal{M}(K)$ be the space of probability measures on $K$ with its weak topology. For a measure $\mu\in\mathcal{M}(K)$, write $d_{\mu}(h)$ for its multifractal spectrum and $E_{\mu}(h)$ for the corresponding level set of points with local dimension $h$. Typical-measure multifractal spectrum conjecture. There exists a constant $0<D<d$ such that, for typical measures $\mu$ in the Baire sense in $\mathcal{M}(K)$, one has $d_{\mu}(h)=h$ for every $h\in[0,D]$, while $E_{\mu}(h)=\emptyset$ whenever $h>D$. This conjecture proposes a universal multifractal spectrum for Baire-typical probability measures on arbitrary compact subsets of Euclidean space, extending the known result for typical measures on $[0,1]^d$; its resolution is not specified in the supplied text.