The seed-and-adjacency conjecture for cluster algebras

Let ${\mathcal A}$ be a cluster algebra of rank $n$, with clusters consisting of $n$ cluster variables. Its exchange graph has clusters as vertices, up to permutation of cluster variables, and adjacency records mutation in one direction. Seed-and-adjacency conjecture. (i) Every seed is uniquely defined by its cluster; thus, the vertices of the exchange graph can be identified with the clusters, up to a permutation of cluster variables. (ii) Two clusters are adjacent in the exchange graph if and only if they have exactly $n-1$ common cluster variables. These assertions describe when seed data are determined by cluster variables alone and characterize exchange-graph adjacency combinatorially. The supplied text gives no resolution status.