Recurrence conjecture for the uniform infinite planar quadrangulation

Let $Q_\infty$ be the uniform infinite planar quadrangulation (UIPQ), rooted at $\rho$, and consider simple random walk on its vertices. A graph is recurrent when simple random walk started at a vertex returns to that vertex almost surely. UIPQ recurrence conjecture. The UIPQ is almost surely recurrent.

The paper explains that recurrence is equivalent to the effective resistance between $\rho$ and infinity being infinite. Establishing this resistance statement remains open in the source, and recurrence of the UIPQ is presented as an open problem.