Chas–Sullivan's free loop space–Hochschild cohomology conjecture

Let $M$ be a simply connected compact oriented smooth manifold, let $LM=\operatorname{map}(S^1,M)$ be its free loop space, and write $\mathbb{H}_*(LM)=H_{*+d}(LM)$ for shifted free loop homology, where $d=\dim M$. Let $S^*(M)$ denote the algebra of singular cochains on $M$, and let $HH^*(S^*(M);S^*(M))$ denote its Hochschild cohomology. Free loop space–Hochschild cohomology conjecture. There is an isomorphism of Gerstenhaber algebras

$$\mathbb{H}_*(LM)\cong HH^*(S^*(M);S^*(M)).$$

This conjecture is the proposed dictionary between string topology and Hochschild cohomology. The source states that it had not yet been proved at the time of writing; the precise general status is not supplied here.