Removal of semipositivity in the taming criterion for almost complex 4-manifolds

Let $(X,J)$ be a closed almost complex $4$-manifold. Suppose there exists a closed $2$-form $\phi$ such that, for every $x\in X$, there is a closed connected pseudoholomorphic curve $C_x$ passing through $x$ with $\int_{C_x}\phi>0$. Semipositivity-removal conjecture. The semipositivity assumption on $\phi$ in Proposition can be removed; under these hypotheses, $(X,J)$ admits a taming symplectic structure. This conjecture asks whether homological positivity of the curves alone suffices for the taming criterion, without requiring the auxiliary closed form to be semipositive.