The density conjecture for the exceptional quivers X6 and X7

Let $X_6$ and $X_7$ denote the exceptional quivers in the classification of mutation-finite quivers. For a quiver $Q$, call it cluster-$\mathbf{g}$-dense if the closure of its cluster $\mathbf{g}$-vector fan is $\mathbb{R}^n$, and half cluster-$\mathbf{g}$-dense if the closures of the cluster $\mathbf{g}$-vector fans of $Q$ and $Q^{op}$ are closed half-spaces in $\mathbb{R}^n$.

Density conjecture for $X_6$ and $X_7$. The quiver $X_6$ is cluster-$\mathbf{g}$-dense, and the quiver $X_7$ is half cluster-$\mathbf{g}$-dense.

These assertions are the expected completion of the near classification of cluster quivers with dense or half-dense $\mathbf{g}$-vector fans. The paper notes partial results: for $X_6$ a cluster cone is the non-positive orthant, while for $X_7$ the cluster fan lies in an open half-space. The conjecture remains open.