Symplectic circle-action conjecture for 4-manifolds

A symplectic 4-manifold is a smooth 4-manifold equipped with a symplectic form, and a circle action is an action of $S^1$ on the manifold. The form and action are symplectic with respect to each other when the circle action preserves the symplectic form.

Symplectic circle-action conjecture. Every symplectic 4-manifold which admits a circle action also admits (possibly a different) symplectic form and circle action which are symplectic with respect to each other.

This asks whether every symplectic 4-manifold carrying any circle action can be equipped with a compatible symplectic circle action. The preceding discussion gives a positive answer for circle actions with fixed points, while the general case is posed as a broader question and is not resolved here.