There is a notion of Poincare-duality for Hochschild homology, which works for $k$-algebras $A$ such that there is a $d\in \mathbf{N}$ 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...).
