Cohomology F-manifold structure on free-loop-space homology

Let $Q$ be a manifold and let $\Lambda Q$ denote its free loop space. The singular homology $\operatorname{H}_*(\Lambda Q)$ carries a natural cohomology F-manifold structure whose tangent-space product at the origin is the Chas–Sullivan loop product. There are ring isomorphisms

$$\operatorname{SH}^*(T^*Q)\cong \operatorname{H}_*(\Lambda Q)\cong \operatorname{HH}^*(C^*Q).$$

Free-loop-space F-manifold conjecture. These ring isomorphisms can be lifted to isomorphisms of cohomology F-manifolds. This conjecture extends the known algebraic structures relating symplectic cohomology, string topology, and Hochschild cohomology; the required natural F-manifold structures and compatibility remain to be established.