Zung's spectral-sequence conjecture relating $SFC(\phi,\mathcal{C})$ and $CA(\phi)$

Let $\phi$ be the flow under consideration, let $G_+$ be its associated graph, and let $SFC(\phi,\mathcal{C})$ and $CA(\phi)$ be the chain complexes described above, where $\mathcal{C}$ is a chosen collection of closed orbits relative to which $\phi$ has no perfect fits. A multi-loop is a loop, possibly non-embedded, in $G_+$; a non-embeddedness grading records the non-embeddedness of a multi-loop. Consider a chain complex $C\Omega(\phi)$ generated by all multi-loops of $G_+$, with differential counting strumming of multi-loops and cobordisms of orbits along transverse surfaces. Decompose this differential according to the two filtrations described below. Zung's conjecture. The differential on $C\Omega(\phi)$ admits the two stated filtrations: one by non-embeddedness, with components $d^0_{SFC}$ and $d^n_{SFC}$, and one by surface type, with components $d^0_{CA}$ counting strumming, $d^1_{CA}$ counting Fried pants, and $d^n_{CA}$ counting other transverse surfaces. The spectral sequence from the first filtration collapses at the $E_1$ page and gives $H_*(SFC(\phi,\mathcal{C}))$, while the spectral sequence from the second filtration has $CA(\phi)$ as its $E_1$ page. The first collapse is expected because non-embedded multi-loops have an even number of resolutions, and the second because $d^0_{CA}$ cancels all but one multi-loop for each multi-orbit. This conjecture proposes a chain-level reconciliation of the two complexes: the first filtration explains the relation with $SFC(\phi,\mathcal{C})$, and the second explains the relation with $CA(\phi)$.