Tensor-product geometric Satake conjecture for Coulomb branches

Let $\mathcal M^\nu(\lambda,\mu)$ be the deformation associated with a decomposition $W=W^1\oplus W^2$, let $\lambda^1$ and $\lambda^2$ be the corresponding dominant coweights, and define

$$\mathfrak A_\chi^\nu(\lambda,\mu)=\{x\in\mathcal M^\nu(\lambda,\mu)\mid \lim_{t\to0}\chi(t)x\text{ exists}\}.$$

Let $\mathfrak g^\vee$ be the Langlands dual Kac–Moody Lie algebra. The tensor-product geometric Satake conjecture. The direct sum

$$\bigoplus_\mu H_{\operatorname{top}}(\mathfrak A_\chi^\nu(\lambda,\mu))$$

has the structure of an integrable representation of $\mathfrak g^\vee$ and is isomorphic to the tensor-product representation $V(\lambda^1)\otimes V(\lambda^2)$. This conjecture proposes a geometric realization of tensor products; the source gives no general resolution.