Formality theorem for Lie bialgebras and quantization of coboundary r-matrices

Let $(g,{\delta }_{\mathrm{\hslash }})$ be a Lie bialgebra. Let $(U_{\mathrm{\hslash }}(g),{\mathrm{\Delta }}_{\mathrm{\hslash }})$ a quantization of $(g,{\delta }_{\mathrm{\hslash }})$ through Etingof-Kazhdan functor. We prove the existence of a $L_{\mathrm{\infty }}$-morphism between the Lie algebra $C(\text{g})=\mathrm{\Lambda }(g)$ and the tensor algebra $TU=T(U_{\mathrm{\hslash }}(g)[-1])$ with Lie algebra structure given by the Gerstenhaber bracket. When $(g,{\delta }_{\mathrm{\hslash }},r)$ is a coboundary Lie bialgebra, we deduce from the formality morphism the existence of a quantization $R$ of $r$.