The bounded-degree almost sure period two conjecture for majority dynamics

Let $G$ be an infinite graph of bounded degree, and choose the initial opinions independently and uniformly from $\{-1,1\}$. Under majority dynamics, each node's opinion almost surely converges to a cycle of period at most two. Almost sure period two conjecture. Every bounded-degree graph has the almost sure period two property. The conjecture extends the almost sure period two theorem for unimodular transitive graphs to all bounded-degree graphs. The paper gives an infinite counterexample without bounded degree, while the bounded-degree case is left open.