There is a notion of Poincare-duality for Hochschild homology, which works for $k$-algebras $A$ such that there is an $d\in \mathbf{N}$ with $\mathrm{Ext}^i(A,A^e)$  is zero except for $i=d$, and $\mathrm{Ext}^d(A,A^e)$ is an invertible $A^e$-module (this is called Poincare-Van-den-Bergh duality)[1]

I was wondering if it would be possible to extend this result to Weibel's Hochschild sheaves ([2]) on a scheme $(X,\mathcal{O}_X)$, which are the (Zariski) sheafificaiton of the assignments $U\mapsto \mathrm{HH}_n(\mathcal{O}_X(U))$ (for each $n\geq 0$) (and maybe 'only' in the case that $X$ is smooth over $k$, and in such a way that it gives back van-den-Bergh's result in the affine case).

I'm currently only learning about Hochschild homology, so I'm not trusting myself on anything that I could come up with, but this sounds like some sort of Serre-Duality and feels very 'formal' (but for example I'm not sure how $\mathrm{Ext}^i(\mathcal{HH}_n,\omega_X)$ would look like for the Hochschild sheaf $\mathcal{HH}_n$ on $X$.)
Has this been spelled out before? I would be very happy about any references (or explanations...).




[1]: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.353.5917&rep=rep1&type=pdf
[2]: https://eudml.org/doc/140240