Gaiotto's factorization-category conjecture for affine Lie superalgebras

Assume $N>M$ and let $c\neq0$ be a non-rational number. Let $\Gr_{\GL_N}$ denote the affine Grassmannian, $U_{M,N}(\bF)$ the relevant unipotent subgroup with character $\chi_{M,N}$, and $\KL_c(\widehat{\mathfrak{gl}}(M|N))$ the category of $G_{M,N}(\bO)$-equivariant objects in $\widehat{\mathfrak{gl}}(M|N)_c$-mod. Gaiotto's factorization-category conjecture. The categories

$$D_{1/c}^{(\GL(M,\bO)\ltimes U_{M,N}(\bF),\chi_{M,N})}(\Gr_{\GL_N})$$

and

$$\KL_c(\widehat{\mathfrak{gl}}(M|N))$$

are equivalent as factorization categories. Similarly,

$$D_{1/c}^{\GL(N,\bO)}(\Gr_{\GL_N}\times\bV)\simeq\KL_c(\widehat{\mathfrak{gl}}(N|N)).$$

These are precise invariant-theoretic forms of the proposed local geometric Langlands duality for the affine Lie superalgebras; the source explicitly notes that the underlying local geometric Langlands statements are not presently known. Its resolution is not supplied.