Menichi's string topology compatibility conjecture for classifying spaces

Let $G$) be a connected compact Lie group of dimension $d$, let $LBG$ denote the free loop space of its classifying space, and let $S_*(G)$ and $S^*(G)$ denote the singular chains and cochains on $G$. Write

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

for the Burghelea–Fiedorowicz–Goodwillie isomorphism, and

$$\mathbb{D}:HH^{*+d}(S_*(G);S^*(G))\buildrel\cong\over\rightarrow HH^*(S_*(G);S_*(G))$$

for the duality isomorphism. Also write

$$H_{S^1}^*(LBG)\buildrel\cong\over\rightarrow HC^*(S_*(G))$$

for the Burghelea–Fiedorowicz–Goodwillie isomorphism in equivariant cohomology, and let the brackets and algebra structures be those specified in the cited results. String topology compatibility conjecture. (i) The composite of the first two isomorphisms is a morphism of graded algebras from the algebra structure on $H^{*+d}(LBG)$ to the underlying algebra of the Gerstenhaber algebra $HH^*(S_*(G);S_*(G))$. (ii) The equivariant isomorphism is a morphism of graded Lie algebras between the specified Lie bracket on $H_{S^1}^*(LBG)$ and the specified Lie bracket on $HC^*(S_*(G))$. These assertions identify the string-topological algebraic structures on the classifying-space loop space with the corresponding Hochschild and cyclic structures; their resolution is not supplied in the source context.