The inverse-limit homeomorphism conjecture for substitution complexes

Let $\mathcal{M}$ be the class of maps under consideration, let $\operatorname{invlim} f$ denote the inverse limit of $f$, let $\chi_f$ be the substitution associated with $f$, and let $\sim_w$ denote the relevant weak equivalence relation on substitutions. The inverse-limit homeomorphism conjecture. For $f,g\in\mathcal{M}$,

$$\operatorname{invlim} f\simeq\operatorname{invlim} g \quad\text{if and only if}\quad \chi_f\sim_w\chi_g.$$

The preceding theorems establish this equivalence in several important subclasses, while the assertion for arbitrary $f,g\in\mathcal{M}$ is left open.