Pisot Substitution Conjecture for one-dimensional inflation tilings

Let a one-dimensional self-similar inflation tiling have Pisot scaling factor $\lambda$, where $\lambda>1$ is a real algebraic integer whose conjugates have modulus strictly less than one. Let $M$ be its abelianisation, or substitution, matrix.

Pisot Substitution Conjecture. The tiling has pure point spectrum if $M$ has an irreducible characteristic polynomial; equivalently, if the algebraic degree of $\lambda$ equals the dimension of $M$, or the number of tile types.

The conjecture is known in the two-tile case and for certain classes of inflation rules, including rules injective on the first tile and constant on the last tile, as well as $\beta$-substitutions. A general proof is missing, and the source notes that irreducibility is not invariant under topological conjugacy even though spectral type is.