The weekly green path conjecture for cluster mutation classes

Let $\boldsymbol{s}$ be a mutation class, let $v_0,v\in\mathbb{E}\mathrm{xch}_{\boldsymbol{s}}$ be vertices of its exchange graph, and let $F^{\boldsymbol{s}}_{v\to v_0}$ and $C^{\pm\boldsymbol{s}}_{v_0\to v}$ be the associated $F$- and $C$-matrices. A path is called weekly green when it has the green-path property used in the source.

Weekly green path conjecture. There is a weekly green path from $v_0$ to $v$. In particular, the products

$$F^{\boldsymbol{s}}_{v\to v_0}C^{\pm\boldsymbol{s}}_{v_0\to v}$$

are non-positive matrices.

The statement generalizes the preceding verification for $\ell$-Kronecker mutation classes, where $\ell\geq2$, to arbitrary mutation classes. Its status is unresolved in the supplied text.