Frisch–Parisi conjecture for prescribed multifractal spectra

Let $d$ be a positive integer, and let $\mathscr{S}_{d}$ be the set of functions $\sigma:\mathbb{R}\to [0,d]\cup\{-\infty\}$ such that $\sigma$ is concave and continuous, has compact support contained in $(0,+\infty)$, and has maximum $d$. For every $\sigma\in\mathscr{S}_{d}$, there exists a Baire functional space of functions defined on $\mathbb{R}^{d}$.

Frisch–Parisi conjecture. In that Baire functional space, any typical element $f$ obeys some multifractal formalism and satisfies

$$\sigma_f=\sigma.$$

This conjecture concerns the inverse problem of constructing Baire spaces in which typical functions have a prescribed singularity spectrum and satisfy a multifractal formalism. The source describes it as a formulation considered by Jaffard and notes that Jaffard obtained a partial solution, so the full assertion is not established here.