The generalized Weinstein conjecture for eventually singular contact manifolds

Let $M$ be an eventually singular contact manifold, and let $Z$ denote its critical set. An orbit in $M\setminus Z$ is quasi-closed if it is either closed or its $\alpha$- and $\omega$-limit sets belong to $Z$. Generalized Weinstein conjecture. Every Reeb vector field of an eventually singular contact manifold admits at least one quasi-closed orbit. The paper presents this as an unsolved reformulation motivated by examples having generalized singular periodic orbits but no regular or singular periodic orbits outside the critical set.