Exceptional Lie superalgebra F(4) Satake conjecture

Let $e\in\mathfrak{sp}(6)$ be a nilpotent element of Jordan type $(3,3)$, let $\operatorname{SL}(2)$ be a maximal reductive subgroup of its centralizer, and choose an $\mathfrak{sl}_2$-triple $(e,h,f)$. Let $U\subset\operatorname{Sp}(6)$ be the unipotent subgroup with Lie algebra $\mathfrak{sp}(6)_{\leq -1}$, and let $\mathcal{D}\mathcal{W}(\mathbb{F}^{2})_{c^{-1}}^{\operatorname{SL}(2,\mathcal{O})\ltimes U(\mathbb{F}),\mathrm{lc}}$ and $\operatorname{Rep}_q(\operatorname{F}(4))$ be the associated categories, where $q=\exp(\pi\sqrt{-1}/c)$. F(4) Satake conjecture. For $c\notin\mathbb{Q}^{\times}$, these categories are equivalent as braided tensor categories, and the equivalence is compatible with the tautological $t$-structures. The claim proposes a geometric Satake realization of the exceptional Lie superalgebra $\mathfrak{f}(4)$; no resolution is given in the supplied text.