show that the DG operad ${{\bf Ger}}^+_{\infty}$ satisfies the minimal model condition [@Sullivan]. This condition implies that the operad ${{\bf Ger}}^+_{\infty}$ is cofibrant and ${{\bf Ger}}^+_{\infty}$-structures can be transferred across quasi-isomorphisms.

Using Tamarkin’s rigidity (see Theorem \[thm:rigidity\]) for the Hochschild cohomology $HH^{{{\bullet}}}(A,A)$ of the polynomial algebra $A =  {{\mathbb K}}[x^1, \dots, x^d]$ we show that a formality quasi-isomorphism for Hochschild cochains ${C^{\bullet}}(A,A)$ coincides with the corresponding part of the transferred ${{\bf Ger}}^+_{\infty}$-structure on the pair $(HH^{{{\bullet}}}(A,A), A)$.

Furthermore, we show that the operad ${{\bf Ger}}^+_{\infty}$ allows us to establish a necessary and sufficient condition (Theorem \[Teorema\]) for an abstract $L_{\infty}$ morphism $$HH^{{{\bullet}}}(A,A)  {\succ\rightarrow}{C^{\bullet}}(A,A)$$ to be homotopic to an $L_{\infty}$ quasi-isomorphism obtained via Tamarkin’s procedure [@DTT], [@Hinich-pro-Dimu], [@Dima-Proof].

We also show that ${{\bf Ger}}^+_{\infty}$ is a sub-operad of the first sheet $E^1({{\rm SC}})$ of the homology spectral sequence for the topological Swiss Cheese operad considered by A. Voronov in [@Sasha-SC]. Finally we prove that the DG operads ${{\bf Ger}}^+_{\infty}$ and $E^1({{\rm SC}})$ are non-formal.

The paper is organized as follows.

In Section \[sect:prelim\] we fix notation and conventions. In this section we also recall necessary facts about Hochschild cochain complex of an $A_{\infty}$-algebra. In Section \[sect:Ger-plus\] we introduce the operad ${{\bf 