Dimension monotonicity conjecture for directed k-neighbor percolation at fixed k

Let $\theta^{\rm D}(k,d)$ denote the percolation probability of the directed $k$-neighbor graph on $\mathbb{Z}^d$. Fixed-k dimension monotonicity conjecture. For all $k,d\geq 2$ with $k\leq d$,

$$\theta^{\rm D}(k,d+1)\geq\theta^{\rm D}(k,d).$$

The conjecture seeks a coupling between consecutive dimensions while keeping $k$ fixed; the paper explains that its validity would propagate results such as $\theta^{\rm D}(2,2)>0$ to all higher dimensions, and it remains open.