The homological hyperbolicity conjecture for geometric realization

Let $\Phi$ be a substitution with tiling space $\Omega_{\Phi}$, and let $H^1_{hyp}$ be the largest subgroup of $H^1(\Omega_{\Phi};\mathbb Z)$ containing the $\Phi^*$-invariant subgroup $H^1_{\Lambda}$ on which $\Phi^*$ is unimodular and hyperbolic. Let $G':\Omega_{\Phi}\to\mathbb T^{D'}$ be the finite-to-one factor map from the geometric realization theorem, with almost-everywhere fiber cardinality $r'$. The homological hyperbolicity conjecture. If $\Phi$ is hyperbolic and $H^1_{hyp}=H^1(\Omega_{\Phi};\mathbb Z)$, then $r'=1$, so $G'$ is almost everywhere one-to-one. This would identify the geometric realization with an almost-everywhere one-to-one toral model whenever the substitution action is unimodular and hyperbolic on all first cohomology; the source does not provide a resolution.