Magnus conjecture for twisted geodesic flows

Let $M$ be a manifold, let $\sigma$ be a symplectic form on $M$, and let $H$ be the twisted geodesic Hamiltonian on $T^*M$. Magnus conjecture. For every $M$ and any symplectic form $\sigma$, a neighborhood of $M$ in $T^*M$ has finite Hofer--Zehnder capacity. Moreover, almost all levels of $H$ carry periodic orbits. When $\sigma$ is symplectic, every low energy level of $H$ has a contractible periodic orbit. The conjecture would extend finite-capacity and periodic-orbit results from special twisted cotangent bundles to arbitrary manifolds and magnetic forms; the supplied context records partial results under additional assumptions.