Equivariant multiplicity conjecture for flag minors

Let $\mathfrak{g}$ be a Lie algebra of finite simply-laced type, let $N$ be the maximal unipotent subgroup associated with $\mathfrak{g}$, let $\Phi_{+}$ be its set of positive roots, and let $x$ be a flag minor in $\mathbb{C}[N]$. Equivariant multiplicity conjecture. The evaluation of $\overline{D}$ on $x$ is of the form

$$\overline{D}(x)=\frac{1}{\prod_{\beta\in\Phi_{+}}\beta^{n_{\beta}(x)}}$$

where $n_{\beta}(x)$ is a nonnegative integer for every positive root $\beta\in\Phi_{+}$. This conjecture predicts that flag minors are characterized by particularly simple equivariant multiplicities, and is motivated by the analogous formula for strongly homogeneous modules; it is proved in types $A_n$ and $D_4$, while the general simply-laced case remains open.