Strict monotonicity conjecture for X-neighbor percolation

Let $\theta^{\rm X}(k,d)$ denote the percolation probability of the $k$-X-neighbor graph on $\mathbb{Z}^d$. X-neighbor monotonicity conjecture. For fixed $d$, the map $k\mapsto\theta^{\rm X}(k,d)$ is strictly increasing on $k\in\{2,\ldots,d\}$. The conjecture is motivated by the increasing single-edge probability and negative correlations, but the authors are unaware of a coupling between different $k$-XnG models; it remains open.