The polytopes in a Poisson hyperplane tessellation

For a stationary Poisson hyperplane tessellation $X$ in $\mathbb {R}^d$, whose generating hyperplane process has a directional distribution satisfying some mild conditions (which hold in the isotropic case, for example), it was recently shown that with probability one every combinatorial type of a simple $d$-polytope is realized infinitely often by the polytopes of $X$. This result is strengthened here: with probability one, every such combinatorial type appears among the polytopes of $X$ not only infinitely often, but with positive density.