Tsygan's generalized Gerstenhaber formality conjecture for chains

Let $A$ be an associative algebra. Write $C^{\bullet}(A,A)$ for its Hochschild cochain complex and $C_{\bullet}(A,A)$ for its Hochschild chain complex. For a Gerstenhaber algebra $V^{\bullet}$, let $V^{\bullet}[\epsilon]$ be the odd-parameter deformation defined by

$$(a+\epsilon b)(c+\epsilon d)=ac+\epsilon\bigl(bc+(-1)^{|a|}ad+(-1)^{|a|}[a,c]\bigr),$$ $$[a+\epsilon b,c+\epsilon d]=[a,c]+\epsilon\bigl([b,c]+(-1)^{|a|+1}[a,d]\bigr).$$

Tsygan's generalized Gerstenhaber formality conjecture. For any associative algebra $A$, the Hochschild chain complex $C_{\bullet}(A,A)$ is a $G_{\infty}$ module over the $G_{\infty}$ algebra $C^{\bullet}(A,A)[\epsilon]$, and its underlying $L_{\infty}$-module structure over $C^{\bullet}(A,A)$ is given by the action of Hochschild cochains through the operators $L_D$. If this holds, then for a smooth manifold $M$ the Hochschild chains and differential forms acquire corresponding $G_{\infty}$-module structures; the source gives no resolution status.