The Lagrangian PSS stable homotopy equivalence

Lagrangian Floer homology is a Hamiltonian isotopy invariant of Lagrangian pairs in a symplectic manifold that counts pseudo-holomorphic curves with boundary conditions on the Lagrangians, and it may be thought of as an infinite-dimensional version of Morse theory. The classical Lagrangian PSS isomorphism relates the Lagrangian Floer homology of the pair $(L,L)$ to the singular homology of $L$. In this talk we will explain the promotion of this isomorphism to an equivalence of stable homotopy types — the Lagrangian Floer homotopy type of the pair $(L,L)$ is equivalent to the stable homotopy type of $L$. Although this is the expected result, there is no written account of such an equivalence.