We call a symplectic manifold 'monotone' if the cohomology class of its symplectic form is its first Chern class. Fine and Panov conjectured that every monotone Hamiltonian $S^1$-space of dimension 6 is diffeomorphic to a Kähler manifold.

In this talk, recent developements, including work in progress, about this conjecture will be presented, concluding that the conjecture holds under the additional assumption that the $S^1$-action extends to a Hamiltonian $T^2$-action.