Noncommutative Hodge-to-de Rham comparison conjecture

Let $k$ be a ring containing ${\mathbb Q}$, and let $A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}$ be a smooth and proper DG algebra over $k$. Write $K^{st}_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$ for its semitopological $K$-theory, with the ${\mathbb Z}[\beta]$-module structure described in the paper, and write $HP_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$ for its periodic cyclic homology. Let $u$ denote the periodicity map. Noncommutative Hodge-to-de Rham comparison conjecture. There exists a functorial map

$$c:K^{st}_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})\to HP_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$$

such that $c(\beta(\alpha))=u(c(\alpha))$ for every $\alpha\in K^{st}_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$, and such that the induced map

$$K^{st}_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})\otimes_{{\mathbb Z}[\beta]}k[\beta,\beta^{-1}]\to HP_{\:\raisebox{1pt}{\text{\circle*{1.5}}}}(A^{\:\raisebox{3pt}{\text{\circle*{1.5}}}})$$

is an isomorphism. The conjecture supplies a comparison between semitopological $K$-theory and periodic cyclic homology, and is relevant because tensoring semitopological $K$-theory with $k$ gives structures analogous to those on the de Rham cohomology of an algebraic variety. Its resolution status is not specified in the supplied text.