Commutative quantisation conjecture for the algebra ${\mathcal Z}$

Let $\sigma$ be an involution of ${\mathfrak g}$, let $H_1,\ldots,H_l$ be the chosen basic invariants with degrees $d_1,\ldots,d_l$, and write $(H_j)_{(i,d_j-i)}$ for their bihomogeneous components. Let ${\varpi}:{\EuScript S}({\mathfrak g})\to{\EuScript U}({\mathfrak g})$ be the symmetrisation map, and suppose that there is a g.g.s. for $\sigma$. Define $\widehat{\mathcal Z}$ to be the subalgebra generated by $\varpi((H_j)_{(i,d_j-i)})$ for $1\leq i\leq l$ and $0\leq i\leq d_i$. Quantisation conjecture. The algebra $\widehat{\mathcal Z}$ is commutative and

$$\operatorname{gr}(\widehat{\mathcal Z})={\mathcal Z}.$$

This is a proposed quantisation of the Poisson-commutative algebra ${\mathcal Z}$; the source does not provide a resolution of the assertion.