The Weinstein conjecture for periodic orbits on contact-type energy levels

Let $(M,\omega)$ be a symplectic manifold and let $H:M\to\mathbb{R}$ be a Hamiltonian. An energy level is a level set $H^{-1}(c)$, and a hypersurface $Y\subset M$ is of contact type if there is a contact form $\lambda$ on $Y$ such that $\omega|_Y=d\lambda$. Weinstein conjecture. For any Hamiltonian on any symplectic manifold, the Hamiltonian flow has a periodic orbit on any compact energy level of contact type. This is equivalently phrased in terms of Reeb dynamics: every Reeb flow on a closed contact manifold should have a closed orbit. Important cases are known, including contact forms on $S^3$, three-manifolds with nonzero second homotopy, and overtwisted contact structures, but the general statement remains open.