The combined standard-hypothesis and discreteness conjecture for real cluster matrices

Let ${\bf SC}$ be the class of real skew-symmetrizable exchange matrices satisfying the sign-coherent property, and let ${\bf B}(B)$ denote the mutation-equivalence class of $B$. The standard hypothesis is the assertion that every mutation-equivalent matrix satisfies the sign-coherent property; the discreteness conjecture is the assertion that a $c$-vector parallel to a coordinate vector has scalar $\alpha=\pm\sqrt{d_i/d_j}$ for a skew-symmetrizer $D=\mathrm{diag}(d_1,\dots,d_n)$. Combined conjecture. If $B\in{\bf SC}$, then for every $B'\in{\bf B}(B)$, both the standard hypothesis and the discreteness conjecture hold. This packages the two preceding conjectures for all mutation-equivalent initial exchange matrices and is introduced for later technical applications; it remains open.