Brüstle–Dupont–Perotin's interval conjecture for maximal green sequence lengths

Let $Q$ be a cluster quiver, meaning a finite connected quiver without loops or oriented cycles. A maximal green sequence is a green mutation sequence that cannot be extended by another green mutation; write $\mathbf{green}_l(Q)$ for the set of maximal green sequences of length $l$. Brüstle–Dupont–Perotin's interval conjecture. The set

$$\{l\in \mathbb{Z}_{\geq 0}\mid \mathbf{green}_l(Q)\neq\emptyset\}$$

is an interval in $\mathbb{Z}$. This conjecture asserts that every integer between two realized lengths is itself realized by a maximal green sequence. The paper calculates the possible lengths for quivers of type $A$ and of type $\tilde{A}_{n,1}$, while the general statement for cluster quivers remains open.