Kähler realization 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. A Kähler realization conjecture asserts that $(M,\omega)$ is $S^1$-equivariantly symplectomorphic to some Kähler manifold $(X,\omega_X,J)$ with a holomorphic Hamiltonian $S^1$-action. The conjecture asks whether all such manifolds arise from Kähler geometry; the source notes that the corresponding question is known in dimensions two and four, while the existence of a closed monotone symplectic manifold that is not Kähler remains unknown in dimensions six, eight, and ten.