Kontsevich's formality conjecture for cyclic cochains

Let $M$ be a $d$-dimensional manifold with volume form $\Omega\in\Omega^d(M)$. Let $\mathfrak g_S^\Omega=(\mathfrak g_S[v],\delta_\Omega)$, where $\mathfrak g_S=\Gamma(\wedge^{\bullet+1}TM)$, $v$ has degree $2$, and $\delta_\Omega=v\operatorname{div}_\Omega$, with $\operatorname{div}_\Omega$ the divergence operator induced by the de Rham differential. Let $\mathfrak g_G^{\mathrm{cycl}}$ be the differential graded Lie algebra of cyclic Hochschild cochains, namely the cochains satisfying

$$\int_M a_0\varphi(a_1,\dots,a_p)\Omega=(-1)^p\int_M a_p\varphi(a_0,\dots,a_{p-1})\Omega.$$

Kontsevich's cyclic formality conjecture. For each volume form $\Omega\in\Omega^d(M)$ there exists an $L_\infty$-quasi-isomorphism of $L_\infty$-algebras

$$F\colon \mathfrak g_S^\Omega\rightsquigarrow\mathfrak g_G^{\mathrm{cycl}}.$$

This is the cyclic analogue of Kontsevich's formality conjecture, relating polyvector fields with divergence differential to cyclic Hochschild cochains. The source gives no evidence of a resolution, so the conjecture is recorded as open.