Limit g-vectors and prime non-real basis elements for Gr(3,9) and Gr(4,8)

Let limit g-vectors, the braid-group action, stable fixed points, and $\operatorname{ch}(T)$ be as defined for the Grassmannian cluster algebras. Let $T^{(1)}\cup\cdots\cup T^{(r)}$ denote the tableau union, with $r\in\mathbb{Z}_{\ge1}$.

Limit-vector and prime-element conjecture. All limit g-vectors for $\mathbb{C}[\operatorname{Gr}(3,9)]$ and $\mathbb{C}[\operatorname{Gr}(4,8)]$ can be obtained by the braid-group action. Moreover, every prime non-real element in the dual canonical basis of $\mathbb{C}[\operatorname{Gr}(3,9)]$ (respectively, $\mathbb{C}[\operatorname{Gr}(4,8)]$) has the form

$$\operatorname{ch}(T^{(1)}\cup\cdots\cup T^{(r)}),$$

for some $r\in\mathbb{Z}_{\ge1}$ and tableaux $T^{(j)}$ obtained from the corresponding stable fixed points by the braid-group action.

The conjecture aims to describe all limit directions and all prime non-real dual canonical basis elements in the two exceptional Grassmannian cases. The supplied text gives no resolution status.