Non-vanishing uniqueness threshold for hyperbolic Poisson-Voronoi percolation in dimension at least three

We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space $\mathbb{H}^d$ for $d\geq 3$. By recent results of Grebík and Recke and d'Achille et al., this "uniqueness threshold" $p_u(\lambda)$ tends to zero as the intensity $\lambda$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products $\mathbb{H}^{d_1}\times\dots\times\mathbb{H}^{d_k}$ with $k,d_1,\dots,d_k\geq 2$. In contrast, for Poisson-Voronoi percolation on the hyperbolic plane $\mathbb{H}^2$, Benjamini and Schramm have shown that $p_u(\lambda)$ tends to one as $\lambda$ tends to zero, and $p_u(\lambda)>1/2$ for all $\lambda>0$. An unpublished argument of D'Achille and Curien shows that for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$, the uniqueness threshold satisfies $p_u(\lambda)\leq 1/2$ for all $\lambda>0$.
Here we will show that $\inf_{\lambda>0}p_u(\lambda)>0$ for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$. This answers a question of Grebík and Recke.