Thue–Morse Stieltjes continued-fraction minimal-polynomial conjecture

Let $k\geq 2$ be an integer, let $a,b$ be two distinct elements of $\mathbb{F}_{2^k}^{\times}$, and let $\mathbf{u}$ be the $(a,b)$-Thue–Morse sequence. Thue–Morse Stieltjes minimal-polynomial conjecture. The Stieltjes continued fraction $\operatorname{Stiel}(x;\mathbf{u})$ is algebraic over $\mathbb{F}_{2^k}(x)$, with minimal polynomial

$$p_0(x)+p_1(x)y+p_2(x)y^2+p_4(x)y^4,$$

where

$$p_0(x)=\frac{a^{2}b^{4}+b^{6}}{a^{4}}x^{2}+\frac{b^{5}}{a^{5}+a^{4}b},\quad p_1(x)=\frac{ab^{4}+b^{5}}{a^{5}}x+\frac{b^{4}}{a^{5}},$$ $$p_2(x)=\frac{b^{4}}{a^{5}}x+\frac{b^{4}}{a^{6}+a^{5}b},\quad p_4(x)=\frac{b^{4}}{a^{6}+a^{5}b}x^{2}.$$

The conjecture predicts an explicit algebraic relation for a family of Stieltjes continued fractions attached to Thue–Morse sequences over finite fields; its status is not resolved in the supplied source.