Global formality at the $G_\infty$-level

In this paper we prove that the sheaf of $\mathcal L$-polydifferential operators for a locally free Lie algebroid $\mathcal L$ is formal when viewed as a sheaf of $G_\infty$-algebras via Tamarkin's morphism of DG-operads $G_\infty\to B_\infty$.
In an appendix we prove a strengthening of Halbout's globalization result for Tamarkin's local quasi-isomorphism.