Fine–Panov conjecture on monotone Hamiltonian circle actions

Let $(M,\omega)$ be a monotone symplectic manifold of dimension six admitting an effective Hamiltonian $S^1$-action.

Fine–Panov conjecture. $M$ is diffeomorphic to a smooth Fano threefold.

This conjecture concerns whether positive-complexity monotone Hamiltonian symplectic manifolds in the lowest dimension where the general question is unresolved arise, up to diffeomorphism, from smooth Fano threefolds. It remains open in general.