The almost one-to-one coding conjecture for Pisot automorphisms

Let an automorphism be represented by $f(x)$, and call it a Pisot automorphism if $f(x)$ or $f(1/x)$ is, up to a sign, monic and irreducible with a Pisot root $\beta$; this $\beta$ is its associated Pisot number. Let $S_\beta$ denote the symbolic coding associated with the Pisot number $\beta$, and call a coding almost $1$-to-$1$ if it has singleton fibers over a subset of full Haar measure.

Almost one-to-one coding conjecture. A Pisot automorphism with associated Pisot number $\beta$ admits an almost $1$-to-$1$ coding by $S_\beta$.

This generalizes Schmidt's conjecture on codings for Pisot units. The statement is motivated by results of Schmidt, and by Sidorov and Vershik for a wider class of Pisot units including all quadratic units; Sidorov reduced the conjecture to algebraic considerations. Its general status is not specified in the source.