Blanc-Toën Chern character conjecture for smooth proper DG categories

Let $A^{\bullet}$ be a small DG category over $\mathbb{C}$. Its semitopological $K$-theory admits a Bott-periodic complex, and there is a natural functorial Chern character

$$K^{st}_{\bullet}(A^{\bullet})_{\mathbb{Q}}(\beta^{-1})\otimes_{\mathbb{Q}}\mathbb{C}\longrightarrow CP_{\bullet}(A^{\bullet}).$$

Here $\beta$ is the Bott periodicity generator and $CP_{\bullet}(A^{\bullet})$ denotes periodic cyclic homology.

Blanc-Toën conjecture. If $A^{\bullet}$ is smooth and proper, the Chern character map is a quasi-isomorphism.

This predicts that periodic cyclic homology is captured by Bott-periodic semitopological $K$-theory for smooth proper DG categories. The source presents it as an expectation and does not give a general proof.