Pisot substitution conjecture

Let $\sigma$ be a Pisot irreducible substitution, meaning that the characteristic polynomial of its incidence matrix is the minimal polynomial of a Pisot number. Let $(X_{\sigma},S)$ be the associated shift. Pisot substitution conjecture. The shift $(X_{\sigma},S)$ has pure discrete spectrum, meaning that it is isomorphic, in a measure-theoretic sense, to a translation on a compact abelian group. The conjecture concerns when substitutions generate one-dimensional quasicrystals and pure discrete spectrum; it was motivated by the connection between Pisot algebraic properties and quasicrystals, but the supplied text does not establish its resolution.