The curve characterization of real Schur roots

Let $\mathcal{Q}$ be an acyclic quiver, let $P_{\mathcal{Q}}$ be the relevant set of seeds, and for each $\sigma\in P_{\mathcal{Q}}$ let $\Gamma_{\sigma}$ be the set of isotopy classes of $\sigma$-admissible curves $\eta$ without self-intersections. For each $\eta\in\Gamma_{\sigma}$, let $w\in W$ be the reflection such that $\upsilon(\eta)\in R_w$, and let $\beta(\eta)$ be the positive real root corresponding to $w$. The curve characterization of real Schur roots.

$$\{\beta(\eta):\eta\in\bigcup_{\sigma\in P_{\mathcal{Q}}}\Gamma_{\sigma}\}$$

is precisely the set of real Schur roots for $\mathcal{Q}$. This gives a geometric characterization of real Schur roots through non-self-intersecting admissible curves. The supplied text does not state whether this claim has been proved or remains open.