The equality of scaling, geometric and symbolic multifractal spectra

Let $(\mathbf{\Phi},\mathbf{p})$ be a weighted self-similar system, let $\mu$ be its associated self-similar measure, and let $[t_{\min},t_{\max}]$ be the interval of relevant scaling regularity values. Write $f(\alpha)$ for the scaling multifractal spectrum, $\hat{f}(t)$ for its concave envelope on $[t_{\min},t_{\max}]$, and $f_g$ and $f_s$ for the geometric and symbolic Hausdorff multifractal spectra, respectively.

Spectrum equality conjecture. For a self-similar measure $\mu$ and all $t\in[t_{\min},t_{\max}]$,

$$\hat{f}(t)=f_g(t)=f_s(t).$$

This conjecture concerns the agreement of the scaling, geometric and symbolic approaches to the multifractal analysis of self-similar measures. It was originally stated in a similar but more restrictive setting in Conjecture 5.8 of the cited work, and is not addressed further in the present paper.