Conjecture that cohomology free vector fields are toral Diophantine flows

Let $X$ be a vector field on a manifold $M$. A smooth function is a cocycle for $X$, and it is a coboundary if there is a smooth function $u$ satisfying

$$\mathcal{L}_Xu=\phi.$$

The vector field $X$ is cohomology free if every smooth cocycle is cohomologous to a constant cocycle. A vector $\alpha\in\mathbb{R}^N$ is Diophantine if there exist constants $c>0$ and $\gamma>0$ such that

$$|n\cdot\alpha|\geq\frac{c}{|n|^{\gamma}}$$

for every $n\in\mathbb{Z}^N\setminus\{0\}$; it defines the constant vector field $X_\alpha\equiv\alpha$ on $\mathbb{T}^N=\mathbb{R}^N/\mathbb{Z}^N$.

Cohomology-free vector field conjecture. Any cohomology free vector field is smoothly conjugated to a Diophantine vector field on a torus.

Constant Diophantine vector fields on tori are the known examples of cohomology free vector fields, and the conjecture asserts that these are the only examples up to smooth conjugacy.