The essential cohomology conjecture for geometric realization

Let $\Phi$ be a substitution with tiling space $\Omega_{\Phi}$. Let $H^1_{hyp}(\Omega_{\Phi};\mathbb Z)$ be the hyperbolic part of first cohomology, and let $H^1_{ess}(\Omega_{\Phi};\mathbb Z)$ be the essential cohomology, defined by intersecting the cohomological images associated with substitutions compatible with $\Phi$. Let $G':\Omega_{\Phi}\to\mathbb T^{D'}$ be the geometric realization map. The essential cohomology conjecture. If $\Phi$ is unimodular and hyperbolic and

$$H^1_{hyp}(\Omega_{\Phi};\mathbb Z)=H^1_{ess}(\Omega_{\Phi};\mathbb Z),$$

then $G'$ is almost everywhere one-to-one. The conjecture strengthens the preceding homological criterion by replacing all first cohomology with essential cohomology; its status is not resolved in the supplied text.