The traditional Pisot substitution conjecture

Let $\Phi$ be a one-dimensional substitution with unimodular irreducible incidence matrix $M$, meaning that the characteristic polynomial of $M$ is irreducible over $\mathbb Q$, and let $\lambda$ be its inflation, assumed to be a Pisot number. The traditional Pisot substitution conjecture. The $\mathbb R$-action on the tiling space $\Omega_{\Phi}$ has pure discrete spectrum. This is the classical one-dimensional Pisot substitution conjecture, which the paper relates to a strengthening involving essential cohomology; no resolution is supplied in the source.