My question concerns the degeneration of the spectral sequence computing Hochschild homology of differential operators on a smooth affine variety $X$.

The spectral sequence arises from the filtration by order of the differential operator and so has $E^{1}$ page $HH(\mathcal{O}(T^{*}X))$. The differentials on this page are well known to be given by $L_{\pi}$, where $\pi$ is the (symplectic) Poisson form on $T^{*}X$, and $L$ denotes Lie derivative wrt a vector field. The spectral sequence degenerates at the second page.

Here is one (very much sketchy) proof - compute the Hochschild homology of $D_{X}$ using general D-module 6-functors formalism. Compute Poisson cohomology of a cotangent bundle explicitly using the symplectic version of the Hodge star operator. Observe they're the same.

Here is how a proof could go in a complex analytic setting; do the computation for a disc and then use a cover by such and a Cech type argument.

It's this second that I'd like an algebraic version of. If we let $\Delta$ denote the formal $n$-disc, and $D_{\Delta}$ denote the algebra of differential operators on this. It's not hard to see that all homology of the $E^{1}$ page of the above spectral sequence for $D_{\Delta}$ is concentrated in degree $2n$, hence degeneration. I'd like to use some Gelfand-Kazhdan formal geometry type argument to conclude degeneration for an arbitrary smooth $X$ but I can't see how to do it. A reference or explanation would be great!

Edit: cf Ben Zvi's answer here for another less sketchy sketch of the d module argument. Microlocalizing Hochschild homology