Identify the microlocal long exact sequence with the Gysin exact sequence

Determine whether the long exact sequence H^*(M; or_M)[−n] → H^*(M) → H^*(S^*M) → +1, arising from the microlocal computation in the proof of Lemma 3.2 via the object μhom and the convolution with sheaf kernels, coincides with the classical Gysin exact sequence associated to the unit sphere cotangent bundle S^*M → M.

Background

In proving a non-positivity result for a spectral invariant c(1; id, φ^H_1), the paper derives a long exact sequence relating cohomologies of M and the unit sphere cotangent bundle S^*M via microlocal sheaf-theoretic techniques. The sequence features terms H^*(M; or_M)[−n], H^*(M), and H^*(S^*M).

Although this exact sequence suffices for the proof at hand, the identification of its morphisms with those of the classical Gysin exact sequence is not established. Confirming this identification would deepen the connection between microlocal sheaf theory and classical topology of sphere bundles.