Tsygan's cyclic formality conjecture

Let $M$ be a smooth manifold, let $C C_{-\bullet}^{-}(C^\infty(M))$ denote the negative cyclic chain complex of $C^\infty(M)$, and let $(\Omega^{-\bullet}(M)[[u]],ud)$ be the completed differential-form complex with formal variable $u$. The zeroth Taylor component is required to be the Connes quasi-isomorphism, given by the $u$-linear extension of the HKR map.

Tsygan's cyclic formality conjecture. There exists a natural $\mathbb C[[u]]$-linear quasi-isomorphism of $L_\infty$-modules

$$F\colon CC_{-\bullet}^{-}(C^\infty(M))\rightsquigarrow (\Omega^{-\bullet}(M)[[u]],ud)$$

such that $F_0$ is the Connes quasi-isomorphism, given by the $u$-linear extension of the HKR map.

This is the cyclic or negative-cyclic analogue of Tsygan's formality conjecture. The source does not state a resolution status for this candidate, so it is recorded as open.