Transcendence criteria for the minimal word of the rational base $3/2$

Let $x_{0}$ be a positive integer, let $x_{n}=\lceil 3x_{n-1}/2\rceil$, and let $w_{n}=2x_{n+1}-3x_{n}\in\{0,1\}$ be the associated word studied by Dubickas; for $x_{0}=1$ the orbit is A061419 and $w$ is the minimal word $g_{3/2}$ of the rational base number system of Akiyama, Frougny and Sakarovitc