Fomin–Zelevinsky's exchange-graph conjecture for cluster patterns

Let $\mathbf{\Sigma}=\{\Sigma_t=(\mathbf{x}_t,\mathbf{y}_t,B_t)\mid t\in\mathbb{T}_n\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. An unlabeled cluster is the set $\{x_{1;t},\dots,x_{n;t}\}$, and an unlabeled seed is the equivalence class of labeled seeds obtained by permutations. Let the exchange graph be the quotient graph of $\mathbb{T}_n$ by equality of unlabeled seeds. Fomin–Zelevinsky's exchange-graph conjecture. The following properties hold: (a) each unlabeled cluster uniquely determines an unlabeled seed containing it; (b) the exchange graph depends only on the $B$-pattern $\mathbf{B}$; and (c) two unlabeled clusters are adjacent if and only if they have exactly $n-1$ common $x$-variables. The conjecture is stated as true in the source: all three claims were proved, with full generality attributed to Cao (2018), and claim (c) also follows from (a).