Percolation threshold conjecture for the 2-X-neighbor graph

For $d\in\mathbb{N}$, let $\theta^{\rm X}(2,d)$ denote the percolation probability of the $2$-X-neighbor graph on $\mathbb{Z}^d$. Two-neighbor X-graph conjecture. The $2$-XnG percolates for all $d\geq 3$ but does not percolate for $d=2$. Numerical experiments motivate this distinction, while the paper states that the corresponding cases remain unresolved.