Bidirectional k-neighbor percolation threshold conjecture

Let the $k$-BnG be the bidirectional $k$-neighbor graph on $\mathbb{Z}^d$, and let $\theta^{\rm B}(k,d)$ denote its percolation probability. Bidirectional threshold conjecture. The $k$-BnG percolates in dimension $d$ if and only if

$$k\geq 2\sqrt{d}.$$

The paper proves only broad bounds on the order of the smallest percolating $k$ as $d$ grows, and explicitly leaves the precise threshold open; this conjectural formula is therefore unresolved.