The higher-dimensional subcriticality characterization conjecture for $d53c$-bootstrap percolation

Fix an integer $d\geqslant2$ and let $\mathcal{U}$ be a $d$-dimensional update family. For $u\in S^{d-1}$, define the half-space

$$\mathbb{H}_u^d=\{x\in\mathbb{Z}^d:\langle x,u\rangle<0\}$$

and the stable set

$$\mathcal{S}(\mathcal{U})=\{u\in S^{d-1}:[\mathbb{H}_u^d]=\mathbb{H}_u^d\}.$$

Let $\mu$ be Lebesgue measure on $S^{d-1}$. The family $\mathcal{U}$ is subcritical if $\mu(H\cap\mathcal{S})>0$ for every hemisphere $H\subset S^{d-1}$. The higher-dimensional subcriticality characterization conjecture.

$$p_c(\mathbb{Z}^d,\mathcal{U})>0\quad\Longleftrightarrow\quad\mathcal{U}\text{ is subcritical}.$$

This extends the two-dimensional conjectural picture to higher-dimensional update families. At present, essentially nothing is known about $\mathcal{U}$-bootstrap percolation in higher dimensions, so both directions remain open.