Morris's infinite-fluctuation conjecture for critical two-dimensional Ising dynamics

Let $\mathcal U$ be a critical two-dimensional family, meaning that its stable directions intersect every open semicircle and have finite intersection with some semicircle. Start the associated $\mathcal U$-Ising dynamics from an independent initial configuration with plus-density $p=1/2$. Morris's conjecture. Almost surely, every vertex changes state infinitely many times.

This extends the corresponding two-dimensional result for the nearest-neighbour zero-temperature Glauber dynamics to all critical families. The source presents it as conjectural; no proof of the general statement is given there.