Baldridge's symplectic circle action conjecture for closed 4-manifolds

Let $M$ be a closed $4$-manifold that admits a symplectic form and a smooth circle action. Baldridge's conjecture. $M$ also admits a symplectic circle action, possibly with respect to a different symplectic form. This conjecture asks whether the existence of a smooth circle action on a symplectic closed $4$-manifold can always be made compatible with some symplectic structure; the source presents it as a general conjecture motivating constructions of non-symplectic smooth circle actions.