Gaiotto's conjecture for quantum general linear supergroups

Let $0<K<N-1$ be positive integers, let $c\in\mathbb{C}^{\times}\setminus\mathbb{Q}^{\times}$, and set

$$q=\exp(\pi\sqrt{-1}/c).$$

Let $\mathcal{D}\mathcal{W}(\mathbb{F}^{2K})_{c^{-1}}^{\operatorname{GL}(2K,\mathcal{O})\ltimes U'_{K-1,N}(\mathbb{F}),\mathrm{lc}}$ be the locally compact equivariant category defined using the Heisenberg action, and let $\operatorname{Rep}_q(\operatorname{GL}(K|N))$ be the corresponding quantum supergroup representation dg-category. Gaiotto's conjecture. The categories

$$\mathcal{D}\mathcal{W}(\mathbb{F}^{2K})_{c^{-1}}^{\operatorname{GL}(2K,\mathcal{O})\ltimes U'_{K-1,N}(\mathbb{F}),\mathrm{lc}}\simeq \operatorname{Rep}_q(\operatorname{GL}(K|N))$$

are equivalent as braided tensor categories, compatibly with the tautological $t$-structures. This reformulates the cited Gaiotto conjecture for $\operatorname{GL}(K|N)$ in geometric Satake form; its status is not resolved in the supplied text.