Chataur–Menichi conjecture on the Gerstenhaber algebra of free-loop homology

Let $G$ be a connected compact Lie group of dimension $d$, let $BG$ be its classifying space, and let $LBG$ denote the free loop space of $BG$. Write $S_*(G)$ for the singular chain algebra of $G$ and $HH^*(S_*(G),S_*(G))$ for its Hochschild cohomology. Chataur–Menichi conjecture. There is an isomorphism of Gerstenhaber algebras

$$H^{*+d}(LBG)\buildrel{\cong}\over\rightarrow HH^{*}(S_*(G),S_*(G)).$$

This conjecture identifies the shifted free-loop homology of the classifying space with the Hochschild cohomology of the chain algebra of the group, including their Gerstenhaber structures. The source gives no resolution of the conjecture.