Fine–Panov's diffeomorphism conjecture for six-dimensional positive monotone Hamiltonian manifolds

Let $(M,\omega)$ be a positive monotone symplectic manifold of dimension six admitting a Hamiltonian circle action. A Fano manifold is a compact complex manifold whose anticanonical line bundle is ample. Fine–Panov's conjecture. $M$ is diffeomorphic to a Fano manifold. The claim concerns whether six-dimensional positive monotone Hamiltonian manifolds share the underlying smooth topology of Fano manifolds; the surrounding discussion states that this question is not known in dimensions six, eight, and ten, while higher-complexity cases remain open.