The Hodge–de Rham conjecture for smooth proper dg categories

Let $k$ be a field of characteristic $0$, and let ${\mathcal A}$ be a smooth proper dg category over $k$. For every integer $n\geq 1$, consider the complex

$$C({\mathcal A})\otimes k[u]/(u^n),$$

where $C({\mathcal A})$ is the mixed complex of ${\mathcal A}$ and $u$ has degree $2$. Hodge–de Rham conjecture. The homology of $C({\mathcal A})\otimes k[u]/(u^n)$ is a flat $k[u]/(u^n)$-module for all $n\geq 1$. This conjecture is known for the dg category of perfect complexes on a smooth projective variety and for a finite-dimensional algebra of finite global dimension, but is wide open in the general case.