The hyperwinding family conjecture for simply connected symplectic four-manifolds

Let $M$ be a simply connected symplectic four-manifold with an $S^2$ family of hyper-winding symplectic forms. A hyper-winding family is an $S^2$ family of symplectic forms satisfying the hyperwinding condition described in the source.

Hyperwinding family conjecture. $M$ is diffeomorphic to the underlying smooth manifold of a $K3$ surface, and the hyperwinding family of symplectic forms is homotopic, through $S^2$ families of symplectic forms, to the $S^2$ family of hyperkähler structures of a $K3$ surface.

The conjecture is motivated by the family blowup formula and the fact that the corresponding family invariant recovers the number $24$ of singular nodal fibers in an elliptic $K3$ surface. Its resolution is not established in the supplied text.