Grimmett's conjecture on infinite directed paths in biased square-lattice orientations

For $0 \leq p \leq 1$, let $\vec{\mathbb{Z}}^2(p)$ be the random orientation of the square lattice in which each horizontal edge is oriented rightwards with probability $p$ and leftwards otherwise, and each vertical edge is oriented upwards with probability $p$ and downwards otherwise, independently of all other edges. Grimmett's conjecture. For each $p \neq 1/2$, $\vec{\mathbb{Z}}^2(p)$ almost surely contains an infinite directed path.

This conjecture concerns the existence of infinite directed paths in biased random orientations of the square lattice. The corresponding assertion remains wide open, while the unbiased case $p=1/2$ is excluded from the claim.