Koszul duality conjecture for quantizations of dual hyperspherical varieties

Let $\scrX$ be a hyperspherical $G$-variety and $\scrX^\vee$ its dual hyperspherical $G^\vee$-variety, with quantizations $Q\scrX$ and $Q\scrX^\vee$. Let $\calC^{\scrX,\operatorname{gr}}_{B\operatorname{-equiv}}$ and $\widehat{\calC}^{\scrX,\operatorname{gr}}_{B\operatorname{-equiv}}$ be the graded categories of compact and locally compact $B$-equivariant objects in $Q\scrX\operatorname{-mod}$, and let $\calC^{\scrX^\vee,\operatorname{gr}}_{B^\vee\operatorname{-mon,uni}}$ and $\widehat{\calC}^{\scrX^\vee,\operatorname{gr}}_{B^\vee\operatorname{-mon,uni}}$ be the corresponding graded categories of unipotent $B^\vee$-monodromic objects. Koszul duality conjecture. There are Koszul equivalences

$$\kappa\colon\calC^{\scrX,\operatorname{gr}}_{B\operatorname{-equiv}}\simeq\calC^{\scrX^\vee,\operatorname{gr}}_{B^\vee\operatorname{-mon,uni}}$$

and

$$\widehat\kappa\colon\widehat{\calC}^{\scrX,\operatorname{gr}}_{B\operatorname{-equiv}}\simeq\widehat{\calC}^{\scrX^\vee,\operatorname{gr}}_{B^\vee\operatorname{-mon,uni}}.$$

These equivalences are proposed as a categorical form of duality for hyperspherical varieties; the source does not establish them in general.