Existence of contractible relative periodic orbits for Hamiltonian group actions

Let $(M,\omega)$ be a compact symplectic manifold with a Hamiltonian action of a compact connected Lie group ${\rm G}$ and moment map $\mu:M\to\mathfrak g^*$. Assume hypothesis $(H1)$, and let $H_t=H_{t+1}:M\to\mathbb{R}$ be a time-dependent ${\rm G}$-invariant Hamiltonian. For $\tau\in\mathfrak g$ with $\mu^{-1}(\tau)\ne\emptyset$, a relative periodic orbit is a periodic Hamiltonian trajectory modulo the ${\rm G}$-action. Existence conjecture. For every such Hamiltonian and every such $\tau$, there exists a contractible relative periodic orbit in $\mu^{-1}(\tau)$. The statement concerns periodic dynamics on a moment-map level set and extends the usual existence problem for contractible periodic Hamiltonian orbits; its resolution is not determined by the supplied text.