The typical-cell width convergence conjecture for high-dimensional Poisson–Voronoi cells

Let $\mathcal V_{\text{typ}}$ be the typical cell of a Poisson–Voronoi tessellation in dimension $d$, with intensity $\lambda=\lambda(d)>0$, and let $B$ denote the unit ball in $\mathbb R^d$. Write $\operatorname{width}(\mathcal V_{\text{typ}})$ for its width and $\operatorname{vol}(B)$ for the volume of $B$. Width convergence conjecture. There is a constant $c$ such that

$$\frac{\operatorname{width}(\mathcal V_{\text{typ}})}{\sqrt[d]{\lambda\cdot\operatorname{vol}(B)}}\xrightarrow[d\to\infty]{\mathbb P}c.$$

The paper proves only non-matching constant upper and lower bounds for the normalized width, while the normalized inradius, outradius, diameter and mean width converge in probability to explicit constants. Determining whether the normalized width converges to a constant remains open.