Integrable complex structure conjecture for Hamiltonian symplectic Fano actions

Let $(Z,\omega)$ be a six-dimensional symplectic Fano manifold with a Hamiltonian $S^1$-action. An integrable $S^1$-invariant complex structure compatible with $\omega$ is an integrable complex structure on $Z$ that is preserved by the $S^1$-action and compatible with $\omega$. Integrable complex structure conjecture. There is an integrable $S^1$-invariant complex structure on $Z$ compatible with $\omega$. This conjecture asks whether every such Hamiltonian symplectic Fano six-manifold admits an algebraic-type complex structure; the source presents it as an open direction for classification.