The hypoelliptic damping resonance conjecture for Anosov geodesic flows

Let $M=S^*\Sigma$, where $(\Sigma,g)$ is a compact Riemannian manifold whose geodesic flow is Anosov. Let $\widetilde P_\epsilon$ be the hypoelliptic operator defined in the source by

$$\widetilde P_\epsilon=\frac1iX+i\epsilon\widetilde\Delta_g.$$

Hypoelliptic damping conjecture. The second part of the theorem for the elliptically perturbed operator $P_\epsilon$ holds with $\widetilde P_\epsilon$ in place of $P_\epsilon$. The source states this as a proposed analogue; it does not provide a resolution.