The reducedness conjecture for affine Grassmannian slices

Let $G$ be a complex semisimple group and let $\mu\leq\lambda$ be dominant coweights. Write $\mathcal{G}r_\mu^\lambda=\mathcal{G}r_\mu\cap\overline{\mathcal{G}r^\lambda}$ for the transversal slice, and let $J_\mu^\lambda\subset\mathcal{O}(\mathcal{G}r_\mu)$ be the explicitly defined Poisson ideal whose vanishing locus is $\mathcal{G}r_\mu^\lambda$. Let $\mathcal{X}_\mu^\lambda$ be the possibly non-reduced scheme defined by $J_\mu^\lambda$. Reducedness conjecture. The ideal of $\mathcal{G}r_\mu^\lambda$ is $J_\mu^\lambda$. Equivalently, $\mathcal{X}_\mu^\lambda$ is reduced. The conjecture concerns whether the explicitly defined scheme structure on every transversal slice has no nilpotents. The paper's abstract states that this conjecture is proved in type A, while the supplied statement is presented without a broader resolution.