Characterization of flag minors by equivariant multiplicities

Let $\mathfrak{g}$ be a Lie algebra of finite simply-laced type, let $W$ be its Weyl group, let $w_0$ be the longest element of $W$, let $N$ be the associated maximal unipotent subgroup, and let $\Phi_{+}$ be the set of positive roots. Characterization conjecture. The flag minors are exactly the cluster variables of $\mathbb{C}[N]$ whose image under $\overline{D}$ is of the form

$$\frac{1}{\prod_{\beta\in\Phi_{+}}\beta^{n_{\beta}}}$$

for some family of nonnegative integers $(n_{\beta})_{\beta\in\Phi_{+}}$. This is a stronger version of the equivariant multiplicity conjecture: it asserts not only the stated form for flag minors but also that no other cluster variables have this form. The paper presents it as a conjecture, with no resolution supplied here.