The seed-by-cluster conjecture for cluster algebras

Let $\mathcal{A}=\mathcal{A}(\mathcal{S})$ be a normalized skew-symmetrizable cluster algebra, where $\mathcal{S}$ is a mutation equivalence class of seeds, and let a cluster mean the set of cluster variables in a seed. Two seeds belong to the same mutation equivalence class as prescribed by $\mathcal{S}$. Seed-by-cluster conjecture. Each seed in a cluster algebra is uniquely determined by its cluster. Two such seeds are related by a mutation if and only if the corresponding clusters share all elements but one. This conjecture would allow the exchange graph to be identified with the dual graph of the cluster complex, with adjacency determined by clusters whose intersection has cardinality $n-1$.