**I-** Is the following statement still a conjecture see this article ?

**Conjecture (?)**
Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild cohomology of cochain complexes associated to $M$ is isomorphic as a Gerstenhaber algebra to $H_{\ast+d}(LM)$ the $d$-shifted homology of the free loop space on $M$.

**II-** What are the recent advances in solving the conjecture ?

**III-** Is there a short proof of the following fact: $HH^{\ast}(C^{\ast}(M))$ is isomorphic to $H_{\ast+d}(LM)$ as a graded vector space ?