Total sign-coherence for mutation-equivalent real exchange matrices

Prove that if a real skew-symmetrizable exchange matrix has sign-coherent $C$- and $G$-patterns, then every exchange matrix mutation-equivalent to it also has sign-coherent $C$- and $G$-patterns.

Background

The paper generalizes $C$- and $G$-matrices from integer skew-symmetrizable exchange matrices to real skew-symmetrizable matrices. In the real setting, sign-coherence is not automatic, so the authors define the sign-coherent class consisting of matrices whose associated $C$- and $G$-patterns are sign-coherent.

The conjecture asks whether sign-coherence is preserved throughout an entire mutation-equivalence class once it holds for one initial exchange matrix. The authors note that the analogous assertion is known in the integer case, whereas it remains a conjectural hypothesis for general real matrices. Establishing it would permit the duality, fan, and synchronicity results developed later in the paper to apply uniformly across mutation-equivalent initial matrices.