The critical probability for Voronoi percolation in the hyperbolic plane tends to $1\mathrm{/}2$

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to $1\mathrm{/}2$ as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm.