Homological Pisot Conjecture for one-dimensional inflation tilings

Let a one-dimensional inflation tiling be unimodular and Pisot, with scaling factor $\lambda$. Let its first rational Čech cohomology group be the first Čech cohomology group with rational coefficients.

Homological Pisot Conjecture. The tiling has pure point spectrum if its first rational Čech cohomology group has rank equal to the algebraic degree of $\lambda$.

This is a topological reformulation intended to avoid the failure of characteristic-polynomial irreducibility to be invariant under topological conjugacy. The source presents it as an open Pisot-type conjecture and does not give a general proof.