Arbitrary-regularity approximation conjecture for Lyapunov vector fields

Let $X$ be a smooth vector field satisfying property ${\bf L}$ on a manifold $M$, and let $r$ be arbitrary. Property ${\bf L}$ means that the set of Lyapunov cohomology classes in $H^1(M;\mathbb R)$ is nonempty. Property ${\bf G}$ means the conjunction of genericity, hyperbolicity, Morse–Smale transversality, and nondegeneracy of closed trajectories; property ${\bf EG}$ means exponential growth at every rest point.

Arbitrary-regularity approximation conjecture. In any $C^r$ neighborhood of $X$, there exists a vector field that coincides with $X$ in a neighborhood of $X$ and satisfies ${\bf L}$, ${\bf G}$, and ${\bf EG}$.

The theorem preceding this statement establishes the analogous existence result in any $C^0$ neighborhood. The conjecture asks whether the same conclusion holds with an arbitrary $C^r$ topology, and its resolution is not given in the source.