Equivariant chain-level isomorphism conjecture for broken trajectories

Let $(f,g)$ be a stably Morse-Smale pair, and suppose that the answer to the relevant question about stably Morse-Smale pairs is positive. Then $(f^\epsilon,g)$ is a Morse-Smale pair and determines the usual Thom-Smale-Witten complex $(C^\epsilon,\partial^\epsilon)$. Let $(\boldsymbol{C},\boldsymbol{\partial})$ be the broken-trajectory chain complex, and let $G$ act on both complexes.

Equivariant chain-level isomorphism conjecture. The chain complexes $(C^\epsilon,\partial^\epsilon)$ and $(\boldsymbol{C},\boldsymbol{\partial})$ are isomorphic as $\mathbb{Z}[G]$-complexes.

This is stronger than the conjectured isomorphism of homology groups because it identifies the complexes together with their $G$-actions. It would give a direct chain-level comparison between the ordinary Thom-Smale-Witten construction and the broken-trajectory model; the necessary geometric gluing input is not yet established in the source.