Hochschild homology conjecture for quantizations of symplectic resolutions

Let $X$ be an irreducible affine Poisson variety admitting a symplectic resolution $\widetilde{X}\to X$. Let $A_\hbar$ be a deformation quantization of $\mathcal{O}_X$, and let $\operatorname{gr}$ denote the associated graded object. Hochschild homology conjecture. (i) For every deformation quantization $A_\hbar$ of $\mathcal{O}_X$, the canonical surjection is an isomorphism

$$\mathsf{HP}_0(\mathcal{O}_X)((\hbar))\xrightarrow{\sim}\operatorname{gr}\mathsf{HH}_0(A_\hbar[\hbar^{-1}]).$$

(ii) There is a countable collection of $\hbar$-homogeneous hypersurfaces in $\hbar H^2(\widetilde{X})[[\hbar]]$ such that, for $A_\hbar$ obtained as the global sections of a quantization in the family $\hbar H^2(\widetilde{X})[[\hbar]]$ outside this collection, there is an abstract isomorphism

$$\mathsf{HP}^{DR}_\bullet(X)((\hbar))\cong\operatorname{gr}\mathsf{HH}_\bullet(A_\hbar[\hbar^{-1}]).$$

The conjecture seeks to relate Poisson and Hochschild homology for quantizations of varieties with symplectic resolutions; its proposed generic statement is motivated by the symmetric-power case, while the general relationship remains open.