Which cluster algebras where the existence of maximal green sequences is still unknown?

Question

Maximal green sequences are studied in many papers. For example, Maximal Green Sequences for Cluster Algebras Associated to the n-Torus by Eric Bucher, On Maximal Green Sequences by Thomas Brüstle, Grégoire Dupont, Matthieu Pérotin, Minimal length maximal green sequences by Alexander Garver, Thomas McConville, Khrystyna Serhiyenko, Maximal green sequences for quivers of finite mutation type by Matthew R. Mills, Maximal Green Sequences of Exceptional Finite Mutation Type Quivers by Ahmet I. Seven.

For which cluster algebras is the existence of maximal green sequences still unknown?

Answer 1

First, it may be better to ask which quivers have a maximal green sequence. Or ask which cluster algebras have a initial seed with a maximal green sequence. Greg Muller shows in The existence of a maximal green sequence is not invariant under quiver mutation that the existence of a maximal green sequence is a property of the quiver not the cluster algebra/mutation class of quiver. In this paper an example a two mutation equivalent quivers is given such that one quiver has a maximal green sequence while the other does not.

Second I mentioned this post to Eric Bucher and he suggested the example of quivers coming from Le diagrams/reduced plabic graphs.

In Green-to-Red Sequences for Positroids by Ford and Serhiyenko it is shown that cluster algebras arising from Le diagrams have a green-to-red sequence (a.k.a. reddening sequence). Note that the existence of a green-to-red sequence does only depend on the cluster algebra. Also it is worth noting that these cluster algebra arising from a Le diagram is thought to be the coordinate ring of the corresponding positroid variety. However, the existence of a green-to-red sequence is weaker than the existence of a maximal green sequence. The existence of a maximal green sequence in this case is still open.