Baldridge's fibering conjecture for free circle actions

Let $X$ be a closed symplectic $4$-manifold admitting a free $S^1$-action, and let $M$ be the orbit space of this action. Baldridge's fibering conjecture. Then the quotient manifold $M$ fibers over $S^1$. Baldridge asked this question as a generalization of the product case. The supplied context says that the authors think their method can be adapted to answer it, but that they do not pursue it in the paper; its status is therefore open here.