Shapovalov's homological Duflo conjecture

Let ${\mathfrak g}$ be a Lie algebra, let $S^\bullet({\mathfrak g})$ be its symmetric algebra, let $U({\mathfrak g})$ be its universal enveloping algebra, and let

$$\varphi_D\colon S^\bullet({\mathfrak g})\to U({\mathfrak g})$$

be the Duflo map. Write $H^\bullet({\mathfrak g};-)$ and $H_\bullet({\mathfrak g};-)$ for Lie algebra cohomology and homology, respectively. The canonical actions of cohomology on homology are denoted by $\blacklozenge$ and $\bigstar$.

Shapovalov's homological Duflo conjecture. The map

$$\varphi_{D\bullet}\colon H_\bullet({\mathfrak g};S^\bullet({\mathfrak g}))\to H_\bullet({\mathfrak g};U({\mathfrak g}))$$

induced by $\varphi_D$ is a map of modules from the $H^\bullet({\mathfrak g};S^\bullet({\mathfrak g}))^{\mathrm{opp}}$-module $H_\bullet({\mathfrak g};S({\mathfrak g}))$ to the $H^\bullet({\mathfrak g};U({\mathfrak g}))^{\mathrm{opp}}$-module $H_\bullet({\mathfrak g};U({\mathfrak g}))$. Thus, for $\alpha\in H^\bullet({\mathfrak g};S({\mathfrak g}))$ and $\beta\in H_\bullet({\mathfrak g};S({\mathfrak g}))$,

$$\varphi_{D\bullet}(\alpha\blacklozenge\beta)=\varphi_D^\bullet(\alpha)\bigstar\varphi_{D\bullet}(\beta).$$

For $0$-cohomology, this says that the $\bigl(S^\bullet({\mathfrak g})\bigr)^{{\mathfrak g}}$-module $\bigl(S^\bullet({\mathfrak g})\bigr)_{{\mathfrak g}}$ and the $\bigl(U({\mathfrak g})\bigr)^{{\mathfrak g}}$-module $\bigl(U({\mathfrak g})\bigr)_{{\mathfrak g}}$ are isomorphic via the Duflo map. The source attributes the conjecture to [Sh2], while the paper proves the corresponding cup-product theorem only for $0$-cohomology; its resolution is not specified here.