The braided fusion and real weight functor conjecture for symmetric varieties

Let $K\subset G$ be the symmetric subgroup of a real form $G_{\mathbb R}\subset G$. Write $X({\mathcal K})$ for the corresponding symmetric variety, $G({\mathcal O})$ for the positive-loop group, $G_{\mathbb R}({\mathcal O}_{\mathbb R})$ for the real positive-loop group, and $\operatorname{Gr}_{\mathbb R}$ for the real affine Grassmannian. Let $\star_f$ denote the fusion product, and let $\operatorname{SVect}$ be the tensor category of finite-dimensional super vector spaces. The abelian equivalence

$$\operatorname{Perv}(X({\mathcal K})/G({\mathcal O}))\simeq\operatorname{Perv}(G_{\mathbb R}({\mathcal O}_{\mathbb R})\backslash\operatorname{Gr}_{\mathbb R})$$

is the equivalence referred to below. Braided fusion and real weight functor conjecture. There exists a natural geometric lift of $\star_f$ making $\operatorname{Perv}(X({\mathcal K})/G({\mathcal O}))$ and $\operatorname{Perv}(G_{\mathbb R}({\mathcal O}_{\mathbb R})\backslash\operatorname{Gr}_{\mathbb R})$ braided monoidal categories, and the abelian equivalence upgrades to an equivalence of braided monoidal categories. Moreover, the real weight functors define a fiber functor

$$\omega:\operatorname{Perv}(G_{\mathbb R}({\mathcal O}_{\mathbb R})\backslash\operatorname{Gr}_{\mathbb R})\to\operatorname{SVect}.$$

This would equip the relevant perverse-sheaf categories with braided tensor structures compatible with the local-global equivalence and would provide the super-Tannakian fiber functor expected from the real weight functors. The statement is presented as an expectation to be proved; no resolution is supplied in the source.