In- and out-percolation equivalence conjecture for directed neighbor graphs

Let $\theta^{\rm D}(k,d)$ be the probability of an infinite directed path starting at the origin in the directed $k$-neighbor graph on $\mathbb{Z}^d$, and let $\theta^{\rm D'}(k,d)$ be the probability of an infinite directed path ending at the origin. In–out percolation equivalence conjecture. For all $d\in\mathbb{N}$ and $1\leq k\leq 2d$,

$$\theta^{\rm D'}(k,d)>0\quad\text{if and only if}\quad\theta^{\rm D}(k,d)>0.$$

The paper does not develop methods for in-percolation and explicitly leaves this general assertion unverified, so it remains open.