Conditional distribution conjecture for the Askey–Wilson Markov process

Let $I(A,B,C,D)$ be the parameter interval from Proposition, let $(Y_t)_{t\in I}$ be the Markov process constructed there, and let $\mathcal{F}_{s,u}$ denote the sigma-algebra generated by the process outside the interval $(s,u)$. For $s<t<u$ in $I(A,B,C,D)$, set

$$x=Y_s+\sqrt{Y_s^2-1},\qquad z=Y_u+\sqrt{Y_u^2-1}.$$

Conditional distribution conjecture. The conditional distribution of $Y_t$ given $\mathcal{F}_{s,u}$ is

$$\nu\left(y; \frac{z\sqrt{t}}{\sqrt{u}},\frac{\sqrt{t}}{z\sqrt{u}},\frac{x\sqrt{s}}{\sqrt{t}},\frac{\sqrt{s}}{x\sqrt{t}}\right).$$

This conjectural formula identifies the bridge law of the Markov process with an Askey–Wilson distribution and would give an explicit description of its conditional distributions. The notation $\nu$ and the process are supplied by the surrounding construction; the candidate is marked unresolved in the provided source information.