Berenstein–Geiss–Muller conjecture on maximal green sequences and upper cluster algebras

Let $Q$ be a quiver, and let $\mathcal{A}(Q)$ denote its cluster algebra. A quiver admits a maximal green sequence when it has a finite mutation sequence satisfying the green-vertex condition at every mutation. The upper cluster algebra is the ring of rational functions that are Laurent polynomials in every cluster.

Berenstein–Geiss–Muller conjecture. The following are equivalent:

1. The quiver $Q$ can be mutated to a quiver which admits a maximal green sequence.
2. The cluster algebra $\mathcal{A}(Q)$ is equal to its upper cluster algebra.

This conjecture proposes that equality with the upper cluster algebra is determined by the existence of a maximal green sequence somewhere in the mutation class. The paper later gives a counterexample, so the conjecture is refuted.