sg.symplectic geometry – How much further could the PSS morphism be pushed?

Given a closed symplectic manifold $(X, omega)$, the well known Piunikhin-Salamon-Schwarz morphism identifies the quantum cohomology $QH(X, omega)$ with the Hamiltonian Floer cohomology $HF(X,H)$ for a non-degenerate Hamiltonian $H in C^{infty}(S^1 times X, mathbb{R})$. Moreover, the PSS morphism respects the product structures.

The quantum product defined over $QH(X, omega)$ uses the genus $0$ three-pointed Gromov-Witten invariants. By considering curves in $X$ with arbitrary genera and marked points, and cohomology classes in the Deligne-Mumford space $mathcal{M}_{g,n}$, the cohomology group $H^{*}(X, mathbb{Q})$ actually has a structure of cohomological field theory, see for instance this survey.

Treating the marked points on the curves as small loops, one might wonder that there should be a Hamiltonian version of the cohomological field theory, and a generalization of the PSS morphism might identify these two models. Does such a thing exist?