Independence-inducing binarizations for the three-symbol distribution

Let $X,Y,Z$ have the distribution specified in the source, with $Z$ determined by $X$ and $Y$ according to the displayed table. For $N\geq1$, let $\overline{Y^N}$ be any binary output of a binarization channel applied to $Y^N$, and let $\overline{Z^N}$ be an output obtained by processing $Z^N$. The independence-inducing binarization conjecture. For every $N\geq1$ and every $\overline{Y^N}$, there exists $\overline{Z^N}$ such that

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

This strengthens the preceding bound-secrecy reduction by disallowing Alice from binarizing. The source presents it as a possible route to proving bound secrecy for the displayed distribution.