\chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,1,1,0} & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,1,1,0} \\
 L_8 & \chi ^X{}_{0,0,1} & \chi ^S{}_{0,1,0,1,1} & \chi ^X{}_{0,0,1} & \chi ^S{}_{0,1,0,1,1} \\
 \hline
\end{array}$

Furthermore, as we will show below, not only every $1$D quantum void is distillable, but can as well be asymtotically distilled (by the iterative application of the same wiring) to a PR-box. Notice, that this does not follow directly from the fact that every nonlocal correlation in a $1$D quantum void is distillable. For instance, there are wirings that distill the nonlocality of the correlations but map them out of the $1$D void. However, if we can find a wiring that distill nonlocality keeping the correlation in the $1$D quantum void then we can guarantee violation up to a PR-box.

For example, for the one-dimensional face associated with $L_7$, using the wiring described in table \[1dvoid\_strategies\] we have that $B_0^0=PR$, $B_0^7=PR$, $B_7^0=PR$ and $B_7^7=L_7$. Thus, $$\begin{aligned}
W(p,p)=(1-(1-c_0)^2)PR + (1-c_0)^2L_7\end{aligned}$$ Now, if we wire two boxes of the form $p_m=(1-(1-c_0)^{2^{m-1}})PR + (1-c_0)^{2^{m-1}} L_7$ we get:

$$\begin{aligned}
q=(1-(1-c_0)^{2^m})PR+(1-c_0)^{2^m}L_7\end{aligned}$$

So we can use the principle of induction, to assure that, starting with a box $p=c_0PR+(1-c_0)L_7$, after $n$ interactions of wirings, we have the box $Q_n$: 