Generalized geometric Satake conjecture for quiver varieties

Let $\Gamma$ be a symmetric Kac–Moody type, let $\mathfrak{g}_{\Gamma}$ be its Kac–Moody algebra, and let $\mathfrak{A}(\mathbf v,\mathbf w)$ be the attracting locus in the corresponding Coulomb-branch variety $M_C(\mathbf v,\mathbf w)$ for the chosen determinant character. Write $\varpi_i$ for the fundamental weights and $H^{BM}_{\operatorname{top}}$ for top Borel–Moore homology.

Generalized geometric Satake conjecture. The vector space

$$\bigoplus_{\mathbf v} H^{BM}_{\operatorname{top}}(\mathfrak{A}(\mathbf v,\mathbf w))$$

should carry an action of $\mathfrak{g}_{\Gamma}$ and be isomorphic to the irreducible representation with highest weight $\sum_i w_i\varpi_i$.

This is a proposed extension of geometric Satake to all symmetric Kac–Moody types. It is developed further in affine type A, while the source gives no resolution of the stated claim.