Zero-temperature Glauber dynamics fixation-threshold conjecture

Let $(\xi_t)_{t\geq 0}$ be zero-temperature Glauber dynamics on $\mathbb{Z}^d$, and define

$$\tilde{p}_c(d)=\inf\{p\geq 0:\mathbb{P}_p[\lim_{t\to\infty}\xi_t(0)=1]=1\}.$$

Here $\tilde{p}_c(d)$ is the threshold for fixation at opinion $1$. Fixation-threshold conjecture. For any $d\geq 2$,

$$\tilde{p}_c(d)=\frac{1}{2}.$$

The one-dimensional value is $\tilde{p}_c(1)=1$, while for $d\geq 2$ it is known that $\tilde{p}_c(d)\in(0,1)$ and that $\tilde{p}_c(d)\to\frac{1}{2}$ as $d$ grows. The conjecture remains open.