Symplectic mapping-torus conjecture for reducible fibers

Let $X=Y\rtimes S^1$ be a symplectic $4$-manifold, with $Y$ a reducible $3$-manifold.

Symplectic reducible-fiber conjecture. Then $Y=S^2\times S^1$ and

$$X=S^2\times T^2.$$

This conjecture predicts that a symplectic mapping torus with reducible fiber can only be the product $S^2\times T^2$. The supplied text gives no resolution or partial result, so its status remains open.