Basic harmonic homology conjecture for traversing vector fields

Let $X$ be a connected compact smooth manifold with boundary, and let $v$ be a traversing and boundary generic vector field on $X$. Let $\mathsf{Harm}^\ast_{\mathsf b}(X,v)$ denote the complex of basically harmonic basic forms, with differential $d$, and let $\mathcal T(v)$ be the trajectory space of the $v$-flow.

Basic harmonic homology conjecture. For every degree $k$,

$$H^k\big(\mathsf{Harm}^\ast_{\mathsf b}(X,v),d\big) \approx H^k(X;\mathbb R)\oplus H^{k-1}(X;\mathbb R)$$

and

$$H^k\big(\mathsf{Harm}^\ast_{\mathsf b}(X,v),d\big) \approx H^k(\mathcal T(v);\mathbb R)\oplus H^{k-1}(\mathcal T(v);\mathbb R).$$

The conjecture is based on a theorem asserting that, for traversing vector fields, the manifold and trajectory space are homology equivalent. The stated identification of basic harmonic homology with the displayed cohomology groups remains unresolved in the supplied text.