Orthosymplectic factorization-category equivalence

Let $M$ and $N$ be the parameters of the orthogonal setup, let $\bF$ be the Laurent series field, and let $U_{M,N}(\bF)$ be the unipotent group equipped with the character $\chi_{M,N}:U_{M,N}(\bF)\to\BG_a$. Let $\Gr_{\SO_N}$ be the affine Grassmannian, and for $c\in\BC^\times$ let $D_{c^{-1}}^{\SO(M,\bO)\ltimes U_{M,N}(\bF),\chi_{M,N}}(\Gr_{\SO_N})$ denote the corresponding twisted equivariant derived category of $D$-modules. In the odd and even cases, let $\on{KL}_c(\widehat\osp(2n-2m|2n))$ and $\on{KL}_c(\widehat\osp(2n|2n-2m-2))$ be the derived Kazhdan–Lusztig categories described in the source. Orthosymplectic factorization-category equivalence. For every $c\in\BC^\times$, in the odd case the categories

$$D_{c^{-1}}^{\SO(M,\bO)\ltimes U_{M,N}(\bF),\chi_{M,N}}(\Gr_{\SO_N})\quad\text{and}\quad\on{KL}_c(\widehat\osp(2n-2m|2n))$$

are equivalent as factorization categories; in the even case the categories

$$D_{c^{-1}}^{\SO(M,\bO)\ltimes U_{M,N}(\bF),\chi_{M,N}}(\Gr_{\SO_N})\quad\text{and}\quad\on{KL}_c(\widehat\osp(2n|2n-2m-2))$$

are equivalent as factorization categories. This is a proposed equivalence between twisted equivariant geometric categories on affine Grassmannians and Kazhdan–Lusztig categories for affine orthosymplectic Lie superalgebras. The source provides no resolution status for either case.