The cluster-variable parametrization conjecture for simply-laced cluster algebras

Let $C=\{u_1,\ldots,u_n\}$ be a cluster in a cluster algebra of simply-laced type and rank $n$, with associated quiver $Q_C$ and category $\operatorname{Mod} Q_C$ defined by the shortest-path relations. Let $\operatorname{Ind}(Q_C)$ be the set of isomorphism classes of indecomposable modules, let $V$ be the set of all cluster variables, and denote by $\alpha_i$ the simple module associated to the vertex $i$. Cluster-variable parametrization conjecture. There exists a bijection

$$b:\operatorname{Ind}(Q_C)\longrightarrow V\backslash C,\qquad \alpha\longmapsto w_\alpha,$$

such that

$$w_\alpha=\frac{P(u_1,\ldots,u_n)}{\prod_i u_i^{n_i}},$$

where $P$ is a polynomial prime to $u_i$ for all $i$, and $n_i=n_i(\alpha)$ is the multiplicity of the simple module $\alpha_i$ in the module $\alpha$. This proposes a uniform correspondence between indecomposable representations of the quiver associated to a cluster and the cluster variables outside that cluster, with the denominator exponents determined by simple-module multiplicities.