Lifts of $C_\infty$- and $L_\infty$-morphisms to $G_\infty$-morphisms

Let $\mathfrak{g}_2$ be the Hochschild complex of cochains on $C^\infty(\mathbb{R}^n)$ and let $\mathfrak{g}_1$ be the space of multivector fields on $\mathbb{R}^n$. In this paper we prove that given any $G_\infty$-structure (i.e. Gerstenhaber algebra up to homotopy structure) on $\mathfrak{g}_2$, and any $C_\infty$-morphism $\varphi$ (i.e. morphism of a commutative, associative algebra up to homotopy) between $\mathfrak{g}_1$ and $\mathfrak{g}_2$, there exists a $G_\infty$-morphism $\Phi$between $\mathfrak{g}_1$ and $\mathfrak{g}_2$ that restricts to $\varphi$. We also show that any $L_\infty$-morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a $G_\infty$-morphism, using Tamarkin's method for any $G_\infty$-structure on $\mathfrak{g}_2$. We also show that any two of such $G_\infty$-morphisms are homotopic.