Kontsevich–Soibelman non-commutative Hodge-to-de Rham degeneration conjecture

Let $A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be a DG algebra over a field $K$ of characteristic $0$. It is saturated if it is both compact, meaning perfect as a complex of $K$-modules, and smooth, meaning perfect as a complex of $A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$-bimodules. The associated Hochschild and cyclic homology complexes are denoted by $HH_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$ and $HC_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$, respectively.

Kontsevich–Soibelman conjecture. For any saturated DG algebra $A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ over a field $K$ of characteristic $0$, the Hodge-to-de Rham spectral sequence

$$HH_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})[u^{-1}] \Rightarrow HC_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{circle*{1.5}}}})$$

degenerates at the first term, equivalently

$$HC_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}) \cong HH_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})[u^{-1}].$$

This is a non-commutative analogue of the Hodge-to-de Rham degeneration theorem for smooth proper algebraic varieties; the conjecture concerns degeneration for all saturated DG algebras over characteristic-zero fields.