Fundamental conjectures on exchange graphs and seeds of cluster algebras

Let $n$ be the rank of a cluster algebra. Its seeds, clusters, cluster variables, mutation equivalence, and exchange graph are understood in the usual sense of cluster algebra theory.

Exchange graph and seed conjectures. (a) The exchange graph of a cluster algebra with rank $n$ only depends on the initial exchange matrix; (b) every seed is uniquely determined by its cluster under mutation equivalence; (c) two clusters are adjacent in the exchange graph if and only if they have exactly $n-1$ common cluster variables.

These are fundamental conjectures concerning the combinatorics of cluster algebras. The supplied source describes them as conjectures, while the paper's abstract says that applications of its cluster formula and $D$-matrix pattern solve some fundamental conjectures affirmatively; the supplied material does not identify which of (a)--(c) are resolved.