Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits

Let $(M,\omega )$ be an aspherical symplectic manifold, which is closed or convex. Let $U$ be an open set in $M$, which admits a circle action generated by an autonomous Hamiltonian $H\in C^{\mathrm{\infty }}(U)$, such that each orbit of the circle action is not contractible in $M$. Under these assumptions, we prove that the Hofer-Zehnder capacity of $U$ is bounded by the Hofer norm of $H$. The proof uses a variant of the energy-capacity inequality, which is proved by the theory of action selectors.