Künneth-component fibered-cone conjecture for symplectic products

Let $N$ be a $3$-manifold and let $S^{1} \times N$ admit a symplectic structure $\omega$. Write the Künneth component of $[\omega]$ in $H^{1}(N,\mathbb R)$ as the corresponding class. Künneth-component fibered-cone conjecture. This component can be represented by a nondegenerate $1$-form; equivalently, it lies in a fibered cone. The source calls this a strict version of the preceding conjecture and explains that it would imply the proposed reformulation using monopole classes, but gives no resolution.