  Let $\mathfrak{g}$ be a finite simply-laced type simple Lie algebra.
Baumann-Kamnitzer-Knutson recently defined an algebra morphism $\overline{D}$
on the coordinate ring $\mathbb{C}[N]$ related to Brion's equivariant
multiplicities via the geometric Satake correspondence. This map is known to
take distinguished values on the elements of the MV basis corresponding to
smooth MV cycles, as well as on the elements of the dual canonical basis
corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke
algebras. In this paper we show that when $\mathfrak{g}$ is of type $A_n$ or
$D_4$, the map $\overline{D}$ takes similar distinguished values on the set of
all flag minors of $\mathbb{C}[N]$, raising the question of the smoothness of
the corresponding MV cycles. We also exhibit certain relations between the
values of $\overline{D}$ on flag minors belonging to the same standard seed,
and we show that in any $ADE$ type these relations are preserved under cluster
mutations from one standard seed to another. The proofs of theses results
partly rely on Kang-Kashiwara-Kim-Oh's monoidal categorification of the cluster
structure of $\mathbb{C}[N]$ via representations of quiver Hecke algebras.
