Conjecture on unit-modulus conjugates and representations of zero

Let $\beta>1$ be an algebraic number, and let $W_{\beta,d}$ denote the set of representations of $0$ over the digit alphabet associated with $\beta$ and digit bound $d$. Assume that

$$W_{\beta,\lceil \beta\rceil-1}\neq\{0^\omega\}.$$

Unit-modulus conjugate conjecture. If $\beta$ has a conjugate of modulus $1$, then $W_{\beta,\lceil \beta\rceil-1}$ is not recognizable by a finite automaton.

The conjecture concerns the relationship between conjugates of $\beta$ on the unit circle and finite automata recognizing representations of zero. The surrounding discussion establishes related results for recognizability in different digit ranges, but no resolution of this assertion is given.