Arithmetic conjecture for expected face numbers of Poisson–Voronoi polytopes

Let $d\in\mathbb{N}$ and $k\in\{0,\ldots,d-1\}$. Write $\mathcal{V}_d$ for the Poisson–Voronoi polytope in dimension $d$, and let $\mathbb{E}f_k(\mathcal{V}_d)$ denote its expected number of $k$-dimensional faces. Arithmetic conjecture. If both $d$ and $k$ are odd, then

$$\mathbb{E}f_k(\mathcal{V}_d)=q\pi^{d-k}$$

for some rational number $q$. The claim is presented as a consequence of the preceding conjecture on dual expected internal-angle sums and is supported by the values discussed in the source, but remains unproved there.