Singularity of harmonic measure for the hyperbolic Poisson–Voronoi tessellation

Let $\mathscr{V}^\lambda$ be the hyperbolic Poisson–Voronoi graph, and let $\nu_0$ be the harmonic measure on $S^1$ obtained from the almost-sure limit of simple random walk on $\mathscr{V}^\lambda$. Thus, for Borel $A\subset S^1$, $\nu_0(A)$ is the probability that random walk converges in the Euclidean metric to a point of $A$. Harmonic-measure singularity conjecture. For almost every realization of $\mathscr{V}^\lambda$, the measure $\nu_0$ is singular with respect to Lebesgue measure on $S^1$. This predicts a dimension-drop phenomenon analogous to that for harmonic measure on infinite supercritical Galton–Watson trees, in contrast with the Lebesgue harmonic measure of hyperbolic Brownian motion on the Poincaré disk; it remains open in the paper.