The localization conjecture for partial flag coordinate rings

Let $G$ be a simple simply connected complex algebraic group of simply-laced type $A$, $D$, or $E$, let $J$ be a subset of the simple roots, and let $K$ be its complementary subset. Let $\widetilde{\cal A}_J$ be the lifted cluster algebra and define $\Sigma_J$ to be the multiplicative submonoid generated by

$$\{\Delta_{\varpi_j,\varpi_j}\mid j\in J\text{ and }\varpi_j\text{ is not minuscule}\}.$$

Here $B_K^-$ is the corresponding opposite parabolic subgroup. Localization conjecture. The localizations of $\widetilde{\cal A}_J$ and $\mathbb{C}[B_K^-\backslash G]$ with respect to $\Sigma_J$ are equal. In particular, when every $\varpi_j$ for $j\in J$ is minuscule, $\Sigma_J$ is trivial and the conjecture asserts equality without localization. The supplied text gives no evidence that this conjecture has been resolved.