The Gerstenhaber algebra conjecture for singular chains and free loop homology

Let $G$ be a topological group such that $M$ is a classifying space of $G$, and let $S_*(G)$ denote the normalized singular-chain algebra of $G$. Write $LM=map(S^1,M)$ for the free loop space, and let $d$ be the dimension parameter of $M$. The Gerstenhaber algebra conjecture. There is an isomorphism of Gerstenhaber algebras

$$H_{*+d}(LM)\cong HH^*(S_*(G),S_*(G)).$$

This would identify shifted free loop homology with Hochschild cohomology for the chain algebra of the classifying group. The source presents this as an expected relationship; its status is not resolved in the supplied text.