Weyl-group module isomorphism for dual hyperspherical varieties

Let $G$ be a complex reductive group with Lie algebra $\fg$, let $\fg^\vee$ be the Langlands-dual Lie algebra, and let $Z_\fg=T^*\CB\times_{\CN_\fg}T^*\CB$ be the Steinberg variety. Its top graded Borel–Moore homology $H(Z_\fg)$ is identified with the group algebra $\BC[W_\fg]$ and canonically identified with $H(Z_{\fg^\vee})$. For a hyperspherical $G$-variety $\scrX$ and its dual $\scrX^\vee$, let $\widetilde{\Lambda}_\scrX$ and $\widetilde{\Lambda}_{\scrX^\vee}$ be the corresponding zero moment levels, and let their top graded Borel–Moore homologies carry the convolution action. Weyl-group module conjecture. The $H(Z_\fg)\cong H(Z_{\fg^\vee})$-modules

$$H(\widetilde{\Lambda}_\scrX)\quad\text{and}\quad H(\widetilde{\Lambda}_{\scrX^\vee})$$

are isomorphic. This is proposed as a representation-theoretic refinement of the expected correspondence between irreducible components; no general proof or disproof is given in the source.