Is there a version of Nest-Tsygan theorem for smooth variety

Question

Let $M$ be a smooth Poisson manifold and $\mathcal{O}_\hbar (M)$ be a deformation quantisation of $\mathcal{O} (M)$. Nest-Tsygan theorem says that $$HH_i(\mathcal{O}_\hbar (M)[\hbar^{-1}])\cong H^{2d-i}_{dR}(X,\mathbb{C}((\hbar))),$$ more explicitly, they showed that the Brylinski spectral sequence degenerates on the first page.

Is the corresponding statement true for smooth Poisson varieties over $\mathbb{C}$, what about non-smooth ones.