The converse characterization of cluster-roots for 2-representation-finite algebras

Let $\Lambda$ be a $2$-representation-finite algebra, and let $\Phi$ be its Coxeter transformation. A root is $\Phi$-positive if

$$\Phi^m(x)\in\mathbb{Z}^l_{\geq 0}$$

for every $m\in\mathbb{Z}$. The cluster-root conjecture. Every $\Phi$-positive root is a cluster-root. Equivalently, $\Phi$-positive roots correspond bijectively to the isomorphism classes of cluster-indecomposable modules. The conjecture gives a criterion for identifying cluster-roots; it is proved in the paper for the class of iterated tilted algebras, but remains open in general.