Lifting central invariants of quantized Hamiltonian actions

Let $G$ be a connected reductive group over an algebraically closed field $\mathbb K$ of characteristic 0, $X$ an affine symplectic variety equipped with a Hamiltonian action of $G$. Further, let $*$ be a $G$-invariant Fedosov star-product on $X$ such that the Hamiltonian action is quantized. We establish an isomorphism between the center of the quantum algebra $\mathbb K[X][[\hbar]]^G$ and the algebra of formal power series with coefficients in the Poisson center of $\mathbb K[X]^G$.