Trajectory-space invariance conjecture for basic de Rham cohomology

Let $X$ be a compact connected smooth manifold and let $v$ be a non-vanishing vector field on $X$. Assume that $v$ admits a Lyapunov function and is transversally generic. Let $\mathcal T(v)$ denote the trajectory space of the $v$-flow, equipped with its stratified topological type.

Trajectory-space invariance conjecture. The $v$-basic de Rham cohomology

$$H^\ast_{\mathsf{basic}\,d\mathcal{R}}(X,v)$$

depends only on the stratified topological type of $\mathcal T(v)$.

The conjecture is motivated by invariance results for basic de Rham cohomology under homeomorphisms in the presence of the Basic Hard Lefschetz property. Whether the stated invariance holds under the given Lyapunov and transversal-genericity assumptions is left open in the supplied text.