that $c_0^{(n)}>c_0^{(n-1)}$ for all values of $c_8^{(n-1)}\neq 1$ and $c_0^{(n-1)}\neq 1$, and $\frac{c_0^{(n)}}{c_0^{(n-1)}}=2-c_0^{(n-1)}-c_8^{(n-1)}\rightarrow1$ only when $c_0^{(n-1)}\rightarrow 1$ or $c_8^{(n-1)}\rightarrow 1$, the last one cannot happen by hypothesis. Hence $c_0$ increases monotonically with $n$ implying that $q_n\rightarrow PR$ as $n\rightarrow\infty$.

![Flow map for the distillation in the $2$D quantum void $\{PR,L_1,L_3\}$. By applying successively the wiring strategy shown in Table \[1dvoid\_strategies\] any point in this set is distilled to PR-box. Interestingly, the flow map shows that the wiring always moves the points up to the isotropic line (dashed red line) and then go up to the PR-box following that line. The dark region shows the region of correlations that do not violate Uffink’s inequality (but do so after the distillation protocol). As well, it follows that all correlations in this void trivialize communication complexity.[]{data-label="flow_map"}](2D_L1L3_flow_map_uffink.pdf)

Some $3$D quantum voids are fully distillable
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Consider a generic $3$D quantum void described by the distribution $$p=c_0 PR+ (1-c_0)\left( c_{i}L_{i} +c_{j}L_{j}  +c_{k}L_{k} \right) .$$ where we can order the coefficients as $c_i \geq c_j \geq c_k$. Considering the $3152$ wirings such that $B^0_0=PR$ we have that there is at most one value of $i$ such that $B^0_{i}=B_0^{i}=PR$. The best one can do is to choose 