Orthosymplectic Satake conjecture for quantum orthosymplectic supergroups

Let $n$ and $k$ be nonnegative integers, let $c\in\mathbb{C}^{\times}\setminus\mathbb{Q}^{\times}$, and set

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

Let $\mathcal{D}\mathcal{W}(\mathbb{F}^{2k})_{1/2+c^{-1}}^{\operatorname{Sp}(2k,\mathcal{O})\ltimes U_k(\mathbb{F}),\mathrm{lc}}$ and $\mathcal{D}\mathcal{W}(\mathbb{F}^{2k})_{c^{-1}}^{\operatorname{Sp}(2k,\mathcal{O})\ltimes U_k(\mathbb{F}),\mathrm{lc}}$ be the locally compact equivariant categories defined from the completed Weyl algebra, and let $\operatorname{Rep}_q(\operatorname{SOSp}(2k+1|2n))$ and $\operatorname{Rep}_q(\operatorname{SOSp}(2n+1|2k))$ be the corresponding finite-dimensional quantum-group dg-categories. The orthosymplectic Satake conjecture. The categories

$$\mathcal{D}\mathcal{W}(\mathbb{F}^{2k})_{1/2+c^{-1}}^{\operatorname{Sp}(2k,\mathcal{O})\ltimes U_k(\mathbb{F}),\mathrm{lc}}\simeq \operatorname{Rep}_q(\operatorname{SOSp}(2k+1|2n))$$

and

$$\mathcal{D}\mathcal{W}(\mathbb{F}^{2k})_{c^{-1}}^{\operatorname{Sp}(2k,\mathcal{O})\ltimes U_k(\mathbb{F}),\mathrm{lc}}\simeq \operatorname{Rep}_q(\operatorname{SOSp}(2n+1|2k))$$

are equivalent as braided tensor categories, with the equivalences compatible with the tautological $t$-structures. These are instances of the paper's proposed quantum geometric Satake equivalences for orthosymplectic Lie superalgebras; the statements are conjectural for the irrational parameters specified here.