Dimension monotonicity conjecture for directed k-neighbor percolation

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 $\mathbb{Z}^d$. Dimension monotonicity conjecture. For all $k\geq 1$, whenever $\theta^{\rm D}(k,d)>0$, we also have $\theta^{\rm D}(k,d+1)>0$. This would imply that the case $k=2$ is critical in every dimension; the conjecture is motivated by the unresolved low-dimensional cases and remains open.