Uniqueness of extremal exchangeable pairs supported by positive dual c-vectors

For a finite type exchange matrix, let a positive dual $\boldsymbol{c}$-vector mean a dual $\boldsymbol{c}$-vector whose entries are positive. An extremal exchangeable pair is an extremal pair of exchangeable rays of the cluster fan.

Positive dual c-vector uniqueness conjecture. Any positive dual $\boldsymbol{c}$-vector supports exactly one extremal exchangeable pair.

This conjecture concerns the extremal exchangeable pairs of the cluster fan and asserts their uniqueness for each positive dual $\boldsymbol{c}$-vector. The surrounding results identify non-initial mesh mutations with extremal exchangeable pairs and show that the associated type cone is simplicial, but the stated uniqueness property is not established here.