Kontsevich's Formality Conjecture for the Hochschild complex

Let $A=C^\infty(X)$ be the algebra of smooth functions on a smooth real manifold $X$, and let $C^\bullet=C^\bullet(A,A)[1]$ be the suspended local Hochschild complex, whose cohomology DGLA is $H^\bullet=\Lambda^\bullet TX[1]$ with trivial differential and the Schouten\Nijenhuis bracket. Kontsevich's Formality Conjecture. The Hochschild complex $C^\bullet$ is quasi-isomorphic as a DGLA to its cohomology $H^\bullet$. This would refine the Hochschild\Kostant\Rosenberg theorem by asserting formality of the Hochschild DGLA, a foundational statement in the deformation quantization of Poisson manifolds.