Sign-coherence and normalized c-vectors conjecture for real rank-3 exchange matrices

Let $B \in \mathrm{M}_{3}(\mathbb{R})$ be an initial exchange matrix, and suppose that its $C$-pattern ${\bf C}(B)$ is sign-coherent. Let $\mathcal{T}$ denote the mutation tree, let $B^{\bf w}$ be the exchange matrix obtained at ${\bf w}\in\mathcal{T}$, and let ${\bf C}(B^{\bf w})$ and ${\bf C}((B^{\bf w})^{\top})$ denote the corresponding $C$-patterns. For a $C$-pattern, write ${\bf c}^{\bf u}_{i;B'}$ for its $i$th $c$-vector, and let $d_i$ be the symmetrizing coefficients.

Sign-coherence and normalization conjecture. (a) For every ${\bf w}\in\mathcal{T}$, all $C$-patterns ${\bf C}(B^{\bf w})$ and ${\bf C}((B^{\bf w})^{\top})$ are sign-coherent.

(b) For every ${\bf w}\in\mathcal{T}$, set $B'=B^{\bf w}$. If a $c$-vector ${\bf c}^{\bf u}_{i;B'}$ in ${\bf C}(B')=\{C_{B'}^{\bf u}\}_{{\bf u}\in\mathcal{T}}$ has the form

$${\bf c}^{\bf u}_{i;B'}=\alpha{\bf e}_{j}$$

for some $\alpha\in\mathbb{R}$ and $j\in\{1,\dots,n\}$, then

$$\alpha=\pm\sqrt{d_i d_j^{-1}}.$$

The conjecture concerns real $C$- and $G$-matrices, where sign-coherence is not known in general. The paper states that it has already been proved for the cluster-cyclic rank-$3$ matrices studied there, so this conjecture is solved in the paper's setting, while its general validity remains open.