Kontsevich–Soibelman degeneration conjecture for smooth and compact DG categories

Let $k$ be a field of characteristic zero and let ${\mathcal A}$ be a small DG category over $k$. Write $HH_{\bullet}({\mathcal A})$ and $HC_{\bullet}({\mathcal A})$ for its Hochschild and cyclic homology, and let $u$ be a formal variable of cohomological degree $2$. If ${\mathcal A}$ is smooth and compact, then the spectral sequence

$$E_1=HH_{\bullet}({\mathcal A})\otimes_k (k[u^{\pm 1}]/uk[u])\Rightarrow HC_{\bullet}({\mathcal A})$$

Kontsevich–Soibelman conjecture. This spectral sequence degenerates at the first sheet. The conjecture generalizes the Hodge-to-de Rham degeneration for smooth projective varieties; it is known for DG algebras concentrated in non-negative degrees, while the general case is the subject of the conjecture.