Weyl-equivariance of equivariant multiplicities for MV cycles

Let $\mathbold{G}$ be a complex reductive group, let $\lambda$ be a dominant coweight lying in the coroot lattice, and let $W$ be the Weyl group of $\mathbold{G}$, canonically identified with the Weyl group of its Langlands dual group. Let $\overline{\mathcal{G}r^\lambda}$ be the corresponding affine Grassmannian Schubert variety, let $S_0$ be the Mirković–Vilonen semi-infinite orbit for the zero coweight, and let $e^\mathbold{T}_0(Z)$ denote the equivariant multiplicity at the unique torus-fixed point $0$ of an MV cycle $Z\subset\overline{\mathcal{G}r^\lambda}\cap S_0$. The zero-weight representation space is identified with $H^\mathrm{top}_c(\overline{\mathcal{G}r^\lambda}\cap S_0)$, whose dual is $H_\mathrm{top}(\overline{\mathcal{G}r^\lambda}\cap S_0)$. Weyl-equivariance conjecture. The map

$$H_\mathrm{top}(\overline{\mathcal{G}r^\lambda}\cap S_0)\longrightarrow\operatorname{Frac}(H^\bullet_\mathbold{T}(\mathrm{pt}))$$

defined on the MV basis by

$$[Z]\longmapsto e^\mathbold{T}_0(Z)$$

and extended linearly is $W$-equivariant. This conjecture asserts that the Weyl group action arising from Geometric Satake is compatible with the action on the fraction field of equivariant cohomology through equivariant multiplicities. The authors state that their work proves it in type $A$ and for $\lambda\leq d\omega_1$, while the general case remains open.