Finiteness conjecture for maximal green sequences of acyclic quivers

Let $Q$ be a quiver with no oriented cycles. A maximal green sequence (MGS) is a sequence of green mutations starting with an extended exchange matrix $\tilde{B}={B\over I}$ and ending with all negative $c$-vectors; a mutation is green when its $c$-vector is positive. Finiteness conjecture. For $Q$ a quiver with no oriented cycles, there are only finitely many MGS. This finiteness is known for tame, affine, and wild quivers with at most three vertices, but remains unknown in general.