The flag-complex conjecture for cluster complexes

Let $\mathcal{A}=\mathcal{A}(\mathcal{S})$ be a cluster algebra, let $\mathcal{X}$ be its set of cluster variables, and let the cluster complex be the simplicial complex on $\mathcal{X}$ whose maximal simplices are the clusters. Two cluster variables are compatible if they appear together in some cluster. Flag-complex conjecture. The cluster complex is always a flag complex: it is the clique complex for the compatibility relation on the set of all cluster variables. The assertion says that every pairwise compatible collection of cluster variables belongs to a common cluster, so the complex is determined by its one-skeleton.