Pisot Substitution Conjecture for irreducible unimodular substitutions

Let $\Phi$ be a 1-dimensional substitution. It is irreducible when the characteristic polynomial of its substitution matrix is irreducible over $\mathbb{Q}$, and unimodular when that matrix has determinant $\pm1$. Let $\Omega_{\Phi}$ be the associated tiling space with its natural $\mathbb{R}$-action.

Pisot Substitution Conjecture. If $\Phi$ is a 1-dimensional irreducible, unimodular, Pisot substitution, then the $\mathbb{R}$-action on $\Omega_{\Phi}$ has pure discrete spectrum.

Pure discrete spectrum is equivalent to the associated dynamical system having entirely discrete spectral behavior, and is relevant to understanding factors of Pisot tiling spaces and pure point diffraction. The source presents this as a named conjecture; no resolution is given here.