Discreteness of real c-vectors parallel to coordinate vectors

Determine whether every real sign-coherent exchange matrix with skew-symmetrizer $D=\operatorname{diag}(d_1,\dots,d_n)$ has the property that any $c$-vector of the form ${\bf c}_{i;t}=\alpha {\bf e}_j$ satisfies $\alpha=\pm\sqrt{d_i d_j^{-1}}$.

Background

In ordinary integer cluster algebras, the lengths of $c$-vectors parallel to coordinate directions satisfy a discrete normalization condition. The paper studies the corresponding phenomenon for real $C$- and $G$-patterns and formulates it for matrices in the sign-coherent class.

The conjecture asserts that if a $c$-vector is parallel to a standard basis vector, then its scalar coefficient is determined, up to sign, by the entries of the skew-symmetrizer. The authors explain that this property is straightforward in important integer cases but is nontrivial for general real exchange matrices. They also derive an equivalent formulation in terms of the corresponding $g$-vectors and use the conjecture in proving dual mutation and fan-structure results.