Benjamini's transient triangulation conjecture

Let $G$ be a bounded-degree plane triangulation. Write $G^{1/2}$ for site percolation on $G$ in which each vertex is retained independently with probability $1/2$. The graph $G$ is transient when simple random walk on $G$ is transient. Benjamini's transient triangulation conjecture. If $G$ is transient, then $G^{1/2}$ has an infinite connected component almost surely. This is open even when the probability of coloring a square black is $2/3$, and is known to the authors to be open for every fixed probability in $[1/2,1)$.