Hamiltonianity conjecture for symplectic circle actions with isolated fixed points

Let $(X^6,\omega)$ be a closed symplectic six-manifold equipped with a symplectic $S^1$ action, and suppose that its fixed-point set contains at least one isolated fixed point. Hamiltonianity conjecture. The action is Hamiltonian. The paper proves a special case of this conjecture for certain non-semifree actions; earlier results establish it for semifree actions and, in the six-dimensional case, under an additional condition on a reduced space at a regular level.