Quantized symplectic duality conjecture for Hochschild homology

Let $\mathfrak{N}$ and $\mathfrak{N}^!$ be as in the symplectic duality conjecture, let $A$ be a quantization of $\mathscr{N}$, and let $A_0$ be its central quotient quantizing $\mathfrak{N}$. Let $S$ be the grading torus, let $u$ denote the degree-two generator of $\operatorname{H}^*(BS)\cong\mathbb{C}[u]$, and let $\mathbb{C}_1$ be the one-dimensional $\operatorname{H}^*(BS)$-module annihilated by $u-1$. Quantized symplectic duality conjecture. There exist canonical isomorphisms of filtered vector spaces

$$H\!H_0(A_0)\cong I\!H^*_S(\mathfrak{N}^!)\otimes_{\mathbb{C}[u]}\mathbb{C}_1 \qquad\text{and}\qquad H\!H_0(A)\cong I\!H^*_{S\times T^!}(\mathfrak{N}^!)\otimes_{\mathbb{C}[u]}\mathbb{C}_1,$$

where the second isomorphism is compatible with the module structure over $Z(A)\cong\operatorname{H}^*(BT^!)$. This is presented as the Hochschild-homology analogue of the preceding symplectic-duality conjecture. The assertion remains open in the supplied text.