Exchange graph conjectures for cluster algebras

Let $\mathcal{A}(\mathcal{S})$ be a cluster algebra with seed $(\tilde{\mathbf{x}},\tilde B)$, principal part $B$, rank $n$, clusters, and exchange graph whose vertices are seeds and whose edges are seed mutations. A cluster variable is an element $x$ occurring in some cluster, and a seed is acyclic when its exchange matrix $B$ is acyclic. Exchange graph conjectures. (1) The exchange graph of $\mathcal{A}(\tilde{\mathbf{x}},\tilde B)$ depends only on the principal part $B$ of $\tilde B$. (2) Every seed in $\mathcal{S}$ is uniquely determined by its cluster; hence two clusters are adjacent if and only if their intersection has cardinality $n-1$. (3) For any cluster variable $x$, the seeds whose clusters contain $x$ form a connected subgraph. (4) The seeds whose exchange matrix $B$ is acyclic form a connected subgraph, possibly empty. These statements concern the combinatorial structure of exchange graphs; parts (1)--(3) are known for cluster algebras of finite type and all four are known for classical types, but the general case remains open.