Poisson zero-cell face-probability limit conjecture

Fix $k\in\{2,3,\ldots\}$ and sample $k$ vertices uniformly at random from the vertices of the random polytope $\operatorname{conv}\Pi_{d,1}$. Let $p_k=\frac{\pi^{k/2}}{2^k\Gamma(\frac{k}{2}+1)}$. Face-probability limit conjecture. The probability that the simplex spanned by these vertices is a $(k-1)$-dimensional face of $\operatorname{conv}\Pi_{d,1}$ converges, as $d\to\infty$, to some limit $q_k\in(0,1)$. Moreover, if the concentration conjecture referred to in the source holds, then $p_k=q_k$. This refines the preceding neighborliness speculation by predicting a nontrivial limiting proportion of $k$-tuples that span faces; no resolution is given in the source.