The compatible-seed existence conjecture for monoidal categorifications

Let $\mathcal{A}$ be a cluster algebra and $\mathcal{C}$ an Artinian monoidal categorification of $\mathcal{A}$. Assume that the simple objects of $\mathcal{C}$ are parametrized by a poset $({\bf M},\leq)$ satisfying the decomposition and compatibility assumptions specified in the source. For a seed $\mathcal{S}$ of $\mathcal{A}$, compatibility means that the associated map from Laurent monomials ordered by the dominance order to $({\bf G},\leq)$ is either increasing or decreasing. Compatible-seed existence conjecture. There exists a compatible seed in $\mathcal{A}$. The conjecture is presented as the paper's main conjecture; the source gives examples where compatibility can be checked, but does not state a general proof or disproof.