Eventual sign coherence for quivers with frozen vertices

Let $Q$ be a connected quiver with at least one frozen vertex, and let $\mathbf M$ be a reduced (weakly) balanced and monotone mutation sequence. Write $Q^{(j)}_{\mathbf M}$ for the quiver after the first $j$ mutations, and call it sign-coherent when its mutable-to-frozen arrow data is sign-coherent. Eventual sign coherence conjecture. There exists $T$ such that $Q^{(j)}_{\mathbf M}$ is sign-coherent for all $j>T$. When $Q$ has one frozen vertex $u$, this says that eventually every mutable vertex is adjacent to $u$. This is presented as an equivalent fixed-quiver formulation of Gekhtman and Nakanishi's asymptotic sign coherence conjecture. The general assertion remains open, although the paper notes examples of balanced and monotone sequences that never produce an ice fork.