Limiting-configuration percolation-threshold conjecture for majority dynamics

For majority dynamics on $\mathbb{Z}^2$, let $\eta_\infty$ be the pointwise limiting configuration and define

$$p_c^\infty=\inf\{p\in[0,1]:\mathbb{P}_p[\eta_\infty\text{ percolates}]>0\}.$$

Let $p_c(t)$ be the critical percolation probability at time $t$. Limiting-threshold conjecture.

$$p_c^\infty=\lim_{t\to\infty}p_c(t).$$

It is known that $\frac12\leq p_c^\infty\leq\lim_{t\to\infty}p_c(t)$, but equality is not known. A proof would require suitable spatial correlation decay for the limiting configuration.