The recursive target-value construction for two copies

Let $a_{ij}:=P_{\overline{Y^2}|Y^2}(0,rs)$ for $r,s\in\{1,2,3\}$, and define $\tau_2$ on the indicated set by

$$\tau_2(0i)=\tau(a_{1j},a_{2j},a_{3j}),\quad \tau_2(i0)=\tau(a_{j1},a_{j2},a_{j3}),$$

where $j=\left\lceil i/2\right\rceil$ for $1\leq i\leq6$, and

$$\tau_2(00)=\tau(\tau(a_{11},a_{12},a_{13}),\tau(a_{21},a_{22},a_{23}),\tau(a_{31},a_{32},a_{33})).$$

The two-copy target-value conjecture. There exists a channel $P_{\overline{Z^2}|Z^2}$ such that

$$P(\overline{Y^2}=0\mid X^2,\overline{Z^2}=\overline{z})=\tau_2(z)$$

for every $z$ in the domain of $\tau_2$. This is a proposed extension of the one-dimensional construction; the source gives no resolution of this claim.