The critical-density conjecture for zero-temperature Glauber dynamics on integer lattices

Let $p_c(G)$ be the infimum of the densities $p\in[0,1]$ for which zero-temperature Glauber dynamics on $G$, started from i.i.d. $\operatorname{Bernoulli}(p)$ opinions, converges almost surely to the all-one configuration. Critical-density conjecture. For every integer $d\geq 2$,

$$p_c(\mathbb{Z}^{d})=\frac{1}{2}.$$

This conjecture concerns the threshold for convergence to the all-one state; the paper notes that $p_c(\mathbb{Z}^{d})\in(0,1)$ for $d\geq2$ and that it tends to $1/2$ as $d$ grows, while equality is open.