Metric invariance of the homology of traversally generic differential complexes

Let $M$ be a compact connected smooth $n$-manifold with boundary, let $g$ be a traversally generic Riemannian metric on $M$, and let $\mathsf R$ be a commutative ring. The metric $g$ determines the differential complexes $\mathbf C_\mho^\ast(SM^\circ,g;\mathsf R)$ and $\mathbf C_\mho^\ast(D(SM),g;\mathsf R)$ of free $\mathsf R$-modules, with terms and differentials defined from the indicated relative cohomology constructions and boundary homomorphisms.

Metric-invariance conjecture. The homology groups of these differential complexes depend only on the connected component of the space of traversally generic metrics on $M$ containing $g$.

The complexes package the stratification data associated with the geodesic flow and are intended to provide localized Poincare-duality invariants. The source states the construction and componentwise invariance, but gives no evidence of a resolution, so the claim is recorded as open.