a PR-box
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Consider a generic $1$D quantum void described by the distribution $$p=c_0 PR+ (1-c_0)L_{i}.$$ In this case, the condition simply states that there should exist at least one wiring for which $\tilde{c}_2=1$. Furthermore, this becomes a sufficient condition as well if the terms $B_{i}^{0}$ and $B_{0}^{i}$ are indeed mapped to $PR$. By searching over the $3152$ wirings, we found that for every local point $L_{i}$ there is a strategy doing that (see Table \[1dvoid\_strategies\]). That is, every point in a $1$D quantum void is distillable and the nonlocality of the wired correlation is raised from $NL(p)=c_0$ to $NL(W(p,p))=c_0(2-c_0)$.

$\begin{array}{ |c|c|c|c|c|} 
 \hline
 L_i & x=0 & x=1 & y=0 & y=1 \\ 
 \hline
L_1 & \chi ^S{}_{0,0,0,0,0} & \chi ^X{}_{1,1,1} & \chi ^S{}_{0,0,1,0,0} & \chi ^X{}_{1,1,0} \\
 L_2 & \chi ^S{}_{0,0,1,0,0} & \chi ^X{}_{1,1,0} & \chi ^S{}_{0,0,0,0,0} & \chi ^X{}_{1,1,1} \\
 L_3 & \chi ^S{}_{0,0,0,1,0} & \chi ^X{}_{1,1,0} & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,0,0,0} \\
 L_4 & \chi ^S{}_{0,0,1,1,1} & \chi ^X{}_{1,1,1} & \chi ^X{}_{0,0,1} & \chi ^S{}_{0,1,1,0,0} \\
 L_5 & \chi ^X{}_{0,0,1} & \chi ^S{}_{0,1,1,0,0} & \chi ^S{}_{0,0,1,1,1} & \chi ^X{}_{1,1,1} \\
 L_6 & \chi ^X{}_{0,0,0} & \chi ^S{}_{0,1,0,0,0} & \chi ^S{}_{0,0,0,1,0} & \chi ^X{}_{1,1,0} \\
 L_7 & 