Binarization and conditional independence for the Renner–Wolf candidate distribution

Let $XYZ$ be the distribution displayed in the source, depending on a parameter $a$. The candidate-distribution binarization conjecture. There exists a value $a$ such that, for every binarization channel $P_{\overline{X}|X}$ and $P_{\overline{Y}|Y}$, there exists a channel $P_{\overline{Z}|Z}$ satisfying

$$(\overline{X^N}\perp\!\!\!\perp\overline{Y^N})\mid\overline{Z^N}$$

for every $N\geq1$, where the channels processing $X^N$ and $Y^N$ are binarizations. This is proposed as a route to proving that the displayed distribution has zero secret-key rate despite positive intrinsic information; the source does not resolve the conjecture.