Invariant-measure periodic-orbit conjecture for geodesic flows

Let $(M,g)$ be a compact Riemannian manifold, let $\mu$ be an invariant probability measure for its geodesic flow, and let $H\subset M$ be a smooth hypersurface. Write $S^*H$ for the unit cotangent bundle over $H$, and, for a periodic geodesic $\gamma$, let $\delta_\gamma$ denote its normalized periodic-orbit measure.

Invariant-measure periodic-orbit conjecture. If $\mu(S^*H)>0$, then $\mu|_{S^*H}$ is supported on a union of periodic geodesics $\gamma$ such that $\gamma\cap S^*H$ has positive arc-length measure and

$$\mu|_{S^*H\cap\gamma}\ll\delta_\gamma.$$

The conjecture concerns the dynamical structure of invariant measures charging the unit cotangent bundle over a hypersurface. In dimension two, it would imply that positive mass on $S^*H$ forces $H$ to have positive-measure intersection with a periodic geodesic; the source presents it as an open question.