Tightness of geodesic deviation in the hyperbolic Poisson–Voronoi graph

Let $\mathscr{V}^\lambda$ be the hyperbolic Poisson–Voronoi graph. For $r>0$, consider a geodesic in $\mathscr{V}^\lambda$ connecting the Voronoi cells containing $-r$ and $r$, and let $D_r$ be the graph distance from this geodesic to the cell containing $0$. Geodesic-deviation conjecture. The family of random variables $(D_r)_{r>0}$ is tight. This asks whether long graph geodesics between opposite points of the hyperbolic plane remain within a tight random distance of the cell containing the origin; it is presented as an open qualitative question about how geodesics in the tessellation compare with hyperbolic geodesics.