Cyclic-invariant restriction conjecture for the Hochschild pairing quasi-isomorphism

Let $M$ be a closed oriented C^-manifold, let $\Omega(M)$ denote its de Rham algebra, and let $\Omega(M)^\vee[-\dim M]$ be the shifted dual complex. Write $\mathrm{CH}^{-*}_{\mathrm{cyc}}(\Omega(M),\Omega(M)^\vee[-\dim M])$ for the (weakly) cyclic-invariant negative Hochschild cochain complex, and let $\Theta$ be the quasi-isomorphism appearing in the Hochschild pairing. Cyclic-invariant restriction conjecture. For every such $M$, the quasi-isomorphism restricts to a quasi-isomorphism on cyclic invariants:

$$\Theta^{-1}(\mathrm{CH}^{-*}_{\mathrm{cyc}}(\Omega(M),\Omega(M)^\vee[-\dim M]))\xrightarrow{\simeq}\mathrm{CH}^{-*}_{\mathrm{cyc}}(\Omega(M),\Omega(M)^\vee[-\dim M]).$$

The question arises because the preceding discussion establishes the relevant quasi-isomorphism for simply connected manifolds but does not determine whether the cyclic-invariant subcomplexes are preserved. The author explicitly states that an answer is not known.