The discreteness conjecture for real cluster $c$-vectors

Let $B\in{\bf SC}$ be a real exchange matrix with skew-symmetrizer $D=\mathrm{diag}(d_1,\dots,d_n)$, where ${\bf SC}$ denotes the class of skew-symmetrizable matrices satisfying the sign-coherent property. A $c$-vector ${\bf c}_{i;t}$ is a column vector in the associated $C$-pattern. Discreteness conjecture. If, for some $i,j\in\{1,\dots,n\}$, there is a scalar $\alpha$ such that

$${\bf c}_{i;t}=\alpha{\bf e}_j,$$

then

$$\alpha=\pm\sqrt{\frac{d_i}{d_j}}.$$

This conjecture specifies the possible lengths and signs of $c$-vectors parallel to coordinate vectors. It is automatic in the ordinary integer setting, whereas for real exchange matrices it is a substantive open problem.