The singular Hamiltonian Seifert conjecture

Let $(M,\omega)$ be a $b^m$-symplectic manifold, and let $H\in C^\infty(M)$ be a proper smooth Hamiltonian. A singular periodic orbit is a periodic orbit in the singular Hamiltonian setting, including the singular dynamics associated with the critical hypersurface. Singular Hamiltonian Seifert conjecture. There exists a proper smooth Hamiltonian whose level sets do not contain any singular periodic orbits. This is the singular analogue of the Hamiltonian Seifert problem. The source says that techniques from the cited work of Ginzburg and Gurel may produce examples with no singular periodic orbit, so the claim is presented as an expected counterexample rather than a resolved result.