a wiring such that $\tilde{c}_2= c_i$. To have distillation in whole $3$D void we should have $\tilde{c}_1= c_j+c_k$ thus implying that $\tilde{c}_0= 0$. However, this is achieved only for a subset of the possible $3$D voids given by $(L_1L_2L_3)$, $(L_1L_2L_4)$, $(L_1L_2L_5)$, $(L_1L_2L_6)$, $(L_1L_3L_4)$, $(L_1L_5L_6)$, $(L_2L_3L_4)$, $(L_2L_5L_6)$, $(L_3L_4L_7)$, $(L_3L_4L_8)$, $(L_3L_7L_8)$, $(L_4L_7L_8)$, $(L_5L_6L_7)$, $(L_5L_6L_8)$, $(L_5L_7L_8)$ and $(L_6L_7L_8)$. For all other cases it follows that $\tilde{c}_2= c_i$, $\tilde{c}_1= c_j$ thus implying that $\tilde{c}_0= c_k$, case in which the necessary condition implies that at least the isotropic line where $c_i=c_j=c_k=1/3$ will not be distillable.

A natural question is then whether distillable $3$D voids can all be distilled up to a PR-box. As mentioned before, a sufficient condition to achieve that is that the wiring maps the $3$D void to the same $3$D void. However, by searching over the $3152$ wirings we could not find any with this property. It could be, however, that searching over all wirings (not necessarily maping the $3$D void to another $3$D void) or considering a distillation protocols based on a higher number of copies would achieve that.

No non-signalling face of dimension $4$ or higher is fully distillable
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The argument given above showing that not every $3$D quantum 