Period-doubling Stieltjes continued-fraction transcendence conjecture

Let $J\in\mathbf{F}_4\backslash\{0,1\}$, and let $\mathbf{u}$ be the $(1,J)$-period-doubling sequence. Period-doubling Stieltjes transcendence conjecture. The Stieltjes continued fraction $\operatorname{Stiel}(x;\mathbf{u})$ is transcendental over $\mathbb{F}_2(z)$. The source motivates this claim by observing that period-doubling Stieltjes continued fractions appear to be transcendental, but gives no resolution of this specific assertion.