Katok's cohomology-free vector field conjecture

Let $\mathcal{M}$ be a compact, connected manifold, and let $X$ be a vector field on $\mathcal{M}$. The vector field $X$ is cohomology free if the associated cohomological equation has no obstructions beyond the constants. Katok's conjecture. If $\mathcal{M}$ admits a cohomology-free vector field $X$, then $\mathcal{M}$ is a torus and $X$ is smoothly conjugate to a constant Diophantine vector field. This conjecture concerns the classification of manifolds and flows with cohomology-free dynamics. Within the class of nilflows, Theorem 1.3 of the cited work supports it, but the statement is presented here without a general resolution.