Two-dimensional directed 2-neighbor percolation conjecture

Let $\theta^{\rm D}(k,d)$ denote the probability that the origin belongs to an infinite directed cluster in the directed $k$-neighbor graph on $\mathbb{Z}^d$. Directed 2-neighbor percolation conjecture.

$$\theta^{\rm D}(2,2)>0.$$

This is one of the principal unresolved cases; the authors suggest that a Peierls argument might prove it, but it remains open.