void can be fully distilled can also be extended to non-signalling faces of dimension $4$ or higher. Consider the isotropic line of a $4$-dimensional NS face $$p=c_0 PR+ (1-c_0)/4\left(L_{i} +L_{j}  +L_{k}+L_{m} \right) ,$$ where the $4$ coefficients of the local part are the same and equal to $c_i=1/4$. The necessary criterion for distillation implies that $\tilde{c}_2 > \tilde{c}_0$. However, by restricting to the wirings such that $B^0_0=PR$ and making $\tilde{c}_2=c_i$, the best we can have for this scenario is $\tilde{c}_1 = c_j + c_k$, so necessarily $\tilde{c}_0=c_m$. That is, there is no wiring satisfying the necessary condition for distillation. Clearly, the argument extends to dimensions higher than $4$, thus showing that in any non-signalling face with dimension $4$ or more, at least the isotropic line of that face will not be entirely distillable.

Interestingly, the set $Q^{1}$ (the first level of the NPA hierarchy) can successfully reproduce some of the $4$D voids. At the same time, to our knowledge, there is no known closed set of correlations between $Q^{1}$ and NS-polytope, which may be an indication that the voids reproduced by $Q^{1}$ may be asymptotically distillable to the PR-box, if more than two copies are considered in the distillation protocol. 