The equivariant symplectic Fano conjecture in dimension six

Let $(M,\omega)$ be a $6$-dimensional symplectic Fano manifold, meaning that $c_1(M)=[\omega]$ in $H^2(M,\mathbb{R})$, and suppose that $M$ admits a Hamiltonian $S^1$-action.

Equivariant symplectic Fano conjecture. $M$ is diffeomorphic to a complex projective Fano $3$-fold.

This is a weaker version of the question whether all $6$-dimensional symplectic Fano manifolds admit a compatible complex projective structure. The paper proves substantial restrictions for Hamiltonian $S^1$-actions, including simple connectivity and $c_1c_2(M)=24$, but the asserted diffeomorphism classification remains open.