Drinfeld's smooth categorical compactification conjecture for coherent D-modules

Let $X$ be a separated scheme of finite type over a field $k$ of characteristic zero. Let $D^b_{\operatorname{coh}}(\mathcal D_X)$ denote the bounded derived category of coherent $\mathcal D$-modules on $X$, more precisely its DG enhancement, and let a smooth categorical compactification mean a DG functor from a smooth and proper DG category inducing a localization up to direct summands with kernel generated by one object.

Drinfeld's conjecture. The DG enhancement of $D^b_{\operatorname{coh}}(\mathcal D_X)$ has a smooth categorical compactification.

This conjecture was suggested to the author by V. Drinfeld. It extends the paper's smooth categorical compactification framework from coherent sheaves to coherent D-modules, and the source does not state a resolution.