The essential homological Pisot conjecture

Let $\Phi$ be a unimodular Pisot family substitution with expansion $\Lambda$, tiling space $\Omega_{\Phi}$, and essential cohomology $H^1_{ess}(\Omega_{\Phi};\mathbb Z)$. The essential homological Pisot conjecture. If

$$\operatorname{rank} H^1_{ess}(\Omega_{\Phi};\mathbb Z)=D(\Lambda),$$

then the $\mathbb R^n$-action on $\Omega_{\Phi}$ has pure discrete spectrum. This is described as a slight strengthening of the homological Pisot criterion and is not resolved in the supplied text.