Kählerness conjecture for six-dimensional monotone Hamiltonian circle manifolds

Let $(M,\omega)$ be a six-dimensional closed monotone symplectic manifold equipped with an effective Hamiltonian circle action.

Kählerness conjecture. Then $(M,\omega)$ is $S^1$-equivariantly symplectomorphic to some Kähler manifold $(X,\omega_X,J)$ with some holomorphic Hamiltonian $S^1$-action.

This conjecture asks whether every six-dimensional closed monotone symplectic manifold admitting an effective Hamiltonian circle action is Kähler in an equivariant sense. The general existence question for non-Kähler monotone symplectic manifolds is known to have a counterexample in dimension twelve, while the corresponding question remains open in dimensions six, eight, and ten.