Duflo's conjecture on invariant Poisson and associative centers

Let $G$ be a connected and simply connected Lie group with finite-dimensional Lie algebra $\mathfrak{g}$, and let $\mathfrak{h}\subset\mathfrak{g}$ be a subalgebra with Lie group $H$. Let $\lambda$ be a character of $\mathfrak{h}$, let $\hat{\lambda}$ be an extension of $\lambda$ to $\mathfrak{g}^*$, and set

$$\mathfrak{h}^{\bot}=\\{l\in\mathfrak{g}^*:l(\mathfrak{h})=0\\}.$$

Let $S(\mathfrak{g})$ and $U(\mathfrak{g})$ denote the symmetric and universal enveloping algebras, respectively, with $\mathfrak{h}$-invariant quotients formed using the ideals generated by $Y+\lambda(Y)$ for $Y\in\mathfrak{h}$. Equip the symmetric quotient with its induced Poisson structure, and write $C_{poiss}$ and $C_{ass}$ for the centers of the corresponding Poisson and associative algebras over $\mathbb{R}$. For $H\in\mathfrak{h}$, define

$$\rho(H)=-\frac{1}{2}\operatorname{Tr}\operatorname{ad}_{\mathfrak{g}}(H).$$

Duflo's conjecture. With these notations, there is an algebra isomorphism

$$C_{poiss}\left[(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{h}_{\lambda})^{\mathfrak{h}}\right]\simeq C_{ass}\left[(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{h}_{\lambda+\rho})^{\mathfrak{h}}\right].$$

The conjecture extends the Duflo isomorphism from invariant functions on a Lie algebra to invariant centers associated with a subalgebra and character. The source attributes it to M. Duflo; no resolution is supplied here, so its status is left open.