Cluster-category correspondence conjecture for hereditary algebras

Let $H$ be a finite-dimensional hereditary algebra with quiver $amma$, let $\operatorname{\mathcal A}(H)$ be the corresponding cluster algebra, and let $\operatorname{\mathcal C}$ be the cluster category associated to $H$. Write $\operatorname{ind}\operatorname{\mathcal C}$ for the indecomposable objects of $\operatorname{\mathcal C}$.

Cluster-category correspondence conjecture. There is a one-to-one correspondence between the cluster variables of $\operatorname{\mathcal A}(H)$ and $\operatorname{ind}\operatorname{\mathcal C}$ inducing a one-to-one correspondence between the clusters of $\operatorname{\mathcal A}(H)$ and the basic tilting objects in $\operatorname{\mathcal C}$.

The conjecture is known when $H$ is the path algebra of a simply-laced Dynkin quiver; the general hereditary case is the proposed extension.