The symplectic product conjecture for closed 3-manifolds

Let $N$ be a closed $3$--manifold. The symplectic product conjecture. If $S^1 \times N$ is symplectic, then there exists a $\phi \in H^1(N;\mathbb{Z})$ such that $(N,\phi)$ fibers over $S^1$. This asks whether the converse of Thurston's implication from fiberedness of a 3-manifold to symplecticity of its product with $S^1$ holds; the source presents it as a problem, and no resolution is given here.