Flag-minor conjecture for strict dominant minuscule modules

Let $\mathfrak{g}$ be a simple Lie algebra of finite simply-laced type, let $W$ be its Weyl group, let $N$ be the associated maximal unipotent subgroup, and let $\mathcal{M}in_{0}^{+}$ denote the strict dominant minuscule elements of $W$. For $w\in W$, let $S(w)$ be the associated homogeneous module. Flag-minor conjecture. For every $w\in\mathcal{M}in_{0}^{+}$, the class of $S(w)$ is a flag minor in $\mathbb{C}[N]$. The conjecture is supported by computations in type $D_4$ and would imply that the prime strongly homogeneous modules are precisely those indexed by $\mathcal{M}in_{0}^{+}$; the general simply-laced case remains open.