Universal Type I behavior for asymmetric initial densities

Let $G=(\mathbb{Z},E)\in\mathcal{G}$ be a graph satisfying the shrink property. For an initial spin configuration whose density of one spin is $p\in(0,1)$, consider the corresponding zero-temperature Glauber Ising model $I(G,p)$.

Universal Type I conjecture. The model $I(G,p)$ is of Type $\mathcal{I}$ for every initial density $p\in(0,1)$.

The paper proves the characterization of Type $\mathcal{I}$ behavior under the symmetric Bernoulli initial distribution $p=\frac12$. On the nearest-neighbor lattice $\mathbb{Z}$, universality for all $p\in(0,1)$ is known, but extending the relevant duality or constructing suitable monotone interfaces for general one-dimensional quasi-transitive graphs with interaction range $K>1$ remains open.