Identify the microlocally derived long exact sequence with the Gysin exact sequence

Prove that the long exact sequence H^*(M; or_M)[−n] → H^*(M) → H^*(S^*M) → +1 obtained in the proof of Lemma 3.2 (via the microlocal calculation for a radial Hamiltonian H = f(|ξ|)) coincides with the classical Gysin long exact sequence for the unit cosphere bundle S^*M → M, including the identification of the maps.

Background

In proving Lemma 3.2 (the non-positivity of a certain spectral invariant), the author computes μhom to identify contributions to morphism spaces at two critical values and thereby deduces a long exact sequence relating H^*(M), H^*(M; or_M), and H^*(S^*M).

While the existence of the sequence suffices for the lemma, the precise identification of the morphisms with those in the Gysin exact sequence is left unresolved. Confirming this identification would give a conceptual topological interpretation of the microlocal calculation.