The multifractal spectrum conjecture for self-similar measures

Let $\mu$ be a self-similar measure on $[0,1]$ and let $\mathfrak{P}$ be its natural sequence of partitions. Write $f_{\mathfrak{P}}^{\mu}(\alpha)$ for the multifractal spectrum, let $\hat{f}_{\mathfrak{P}}^{\mu}(t)$ be its concave envelope on $[t_{\min},t_{\max}]$, and let $f_g$, $f_s$, and $b^*$ denote the associated generalized, symbolic, and box-counting spectra, respectively.

Multifractal spectrum conjecture. For all $t \in [t_{\min},t_{\max}]$,

$$\hat{f}_{\mathfrak{P}}^{\mu}(t)=f_g(t)=f_s(t)=b^*(t).$$

The conjecture would identify the concave envelope of the multifractal spectrum with the generalized, symbolic, and box-counting spectra throughout the full regularity range. The supplied context establishes this equality in the special case $\mathbf{p}=(1/3,2/3)$, while the general self-similar-measure statement is presented as a conjecture.