Face connectivity conjecture for cluster algebras

Let ${\mathcal X}$ be a set of cluster variables. A seed is a cluster together with its exchange data, and the exchange graph is the graph whose vertices are seeds and whose edges correspond to seed mutations. Face connectivity conjecture. For any set ${\mathcal X}$ of cluster variables, the seeds whose clusters contain ${\mathcal X}$ as a subset induce a connected subgraph of the exchange graph. The source calls this a stronger conjecture and says it will be proved for some matrices $B$; no general resolution is given.