Transcendence of all Arnoux–Rauzy beta-expansions

Prove that for every Arnoux–Rauzy word $w$ over the alphabet $\{0,1,\dots,d\}$ and every algebraic base $\beta$ with $|\beta|>1$, the number $(w)_\beta=\sum_{i=0}^{\infty}w_i\beta^{-i}$ is transcendental.

Background

Arnoux–Rauzy words generalize Sturmian words to larger alphabets and are among the principal classes considered in the paper. Earlier results established transcendence for certain Arnoux–Rauzy words, including words on a three-letter alphabet, while subsequent work obtained only partial dichotomy results for broader episturmian classes. The paper proves the desired transcendence statement for large subclasses satisfying tiling and recurrence conditions, but leaves the unrestricted question unresolved.