The substitution equivalence conjecture for substitution tiling spaces

Let $\mathcal{T}_{\chi}$ and $\mathcal{T}_{\psi}$ be the tiling spaces associated with substitutions $\chi$ and $\psi$, and let $\tilde{\chi}$ and $\tilde{\psi}$ denote the associated substitutions appearing in the inverse-limit model. Let $\sim_w$ denote the relevant weak equivalence relation. The substitution tiling-space conjecture.

$$\mathcal{T}_{\chi}\simeq\mathcal{T}_{\psi}\quad\text{if and only if}\quad\tilde{\chi}\sim_w\tilde{\psi}.$$

This is the restriction of the preceding general conjecture to substitution tiling spaces. The paper proves the forward implication at the level of the associated matrices and the corresponding Perron-eigenvalue field consequence, but does not establish the asserted equivalence in full.