Categorical quantization commutes with reduction

Let $(X,\omega_X,G,\mathbf J)$ be a Hamiltonian $G$-space for $G$ a reductive group, and let $\mathcal X_{red}$ be the derived Marsden–Weinstein quotient associated with a coadjoint orbit as in the derived Marsden–Weinstein quotient theorem. Let $\mathfrak D^G_{qcoh}(X)$ be a dg-enhancement of the derived category of $G$-equivariant quasi-coherent sheaves on $X$, and let $\mathfrak D_{qcoh}(\mathcal X_{red})$ be a dg-enhancement of the derived category of quasi-coherent sheaves on $\mathcal X_{red}$. Categorical quantization commutes with reduction. There exists an equivalence of dg-categories

$$\mathfrak D^G_{qcoh}(X)_{\omega_X}\simeq \mathfrak D_{qcoh}(\mathcal X_{red})_{\omega_{red}},$$

where the subscript denotes the category deformed in the direction of $\omega_X$ or $\omega_{red}$, respectively. This conjecture expresses the expectation that quantization should commute with symplectic reduction even when the quotient is singular and must be understood as a derived Artin stack; its status is not resolved in the source.