Compatibility conjecture for quantization and reduction via adapted formality

Let $M$ be a Poisson manifold with a Lie algebra $\mathfrak{g}$-action, let $J$ be an equivariant momentum map, and let $M_{\mathrm{red}}$ denote the reduced space. Let $F\colon T_{\mathrm{poly}}(M)\to D_{\mathrm{poly}}(M)$ be a $\mathfrak{g}$-adapted $L_\infty$ morphism, and let $G\colon T_{\mathrm{poly}}(M_{\mathrm{red}})\to D_{\mathrm{poly}}(M_{\mathrm{red}})$ be the $L_\infty$-quasi-isomorphism obtained by the stated zig-zag, with both morphisms constructed using the same Drinfel'd associator. The associated twisted curved $L_\infty$ morphism is

$$F^\mathfrak{g}_J\colon \big(T_\mathfrak{g}(M)[[\hbar]],\hbar\lambda,-[J,\cdot],[\cdot,\cdot]_\mathfrak{g}\big)\longrightarrow \big(D_\mathfrak{g}(M)[[\hbar]],\hbar\lambda,\partial_\mathfrak{g}-[J,\cdot],[\cdot,\cdot]_\mathfrak{g}\big).$$

The reduction maps $T_{\mathrm{red}}$ and $D_{\mathrm{red}}$ fit into a diagram with $G$ whose curved $L_\infty$-morphism square commutes up to homotopy.

Compatibility conjecture. The diagram of curved $L_\infty$-morphisms

$$\begin{CD} \big(T_\mathfrak{g}(M)[[\hbar]],\hbar\lambda,-[J,\cdot],[\cdot,\cdot]_\mathfrak{g}\big) @>{F^\mathfrak{g}_J}>> \big(D_\mathfrak{g}(M)[[\hbar]],\hbar\lambda,\partial_\mathfrak{g}-[J,\cdot],[\cdot,\cdot]_\mathfrak{g}\big)\\ @V{T_{\mathrm{red}}}VV @VV{D_{\mathrm{red}}}V\\ \big(T_{\mathrm{poly}}(M_{\mathrm{red}})[[\hbar]],[\cdot,\cdot]\big) @>>{G}> \big(D_{\mathrm{poly}}(M_{\mathrm{red}})[[\hbar]],\partial,[\cdot,\cdot]\big) \end{CD}$$

commutes up to homotopy.