The conjecture that the partial cluster algebra equals the unipotent coordinate ring

Let $G$ be of simply-laced type $A$, $D$, or $E$, let $J$ be the chosen subset of simple roots, let $K$ be its complementary subset, and let $N_K$ be the corresponding unipotent subgroup. Let ${\cal A}_J$ be the cluster algebra constructed from the associated reduced expressions. Equality conjecture for ${\cal A}_J$. We have

$${\cal A}_J=\mathbb{C}[N_K].$$

This is the unlocalized equality asserted for the cluster algebra associated with the partial flag data. The supplied text gives no evidence that it has been proved or disproved.