The $S$-adic Pisot conjecture

Let $S$ be a finite or infinite set of unimodular substitutions over the finite alphabet $\mathcal{A}$, and let $\boldsymbol{\sigma}\in S^\mathbb{N}$ be a primitive, algebraically irreducible, and recurrent directive sequence with balanced language $\mathcal{L}_{\boldsymbol{\sigma}}$. Let $\mathcal{C}_{\mathbf{1}}$, $\mathbf{1}^{\bot}$, $X_{\boldsymbol{\sigma}}$, $\Sigma$, $\mu$, $d$, and $\mathbb{T}^{d-1}$ denote the associated Rauzy fractal, hyperplane, $S$-adic shift, shift transformation, invariant measure, alphabet dimension, and torus, respectively. The $S$-adic Pisot conjecture. $\mathcal{C}_{\mathbf{1}}$ forms a tiling of $\mathbf{1}^{\bot}$, and the $S$-adic shift $(X_{\boldsymbol{\sigma}},\Sigma,\mu)$ is measurably conjugate to a translation on the torus $\mathbb{T}^{d-1}$; in particular, its measure-theoretic spectrum is purely discrete. The statement extends the well-known Pisot substitution conjecture to the $S$-adic setting, with balancedness providing the analogue of the Pisot hypothesis. Its resolution under the stated hypotheses is not given in the supplied source context.