Foliation adjunction inequality for taut foliations on four-manifolds

Let $M^4$ be a four-manifold with a taut foliation ${\mathcal F}$, and let $J_{\mathcal F}$ be an almost-complex structure induced by ${\mathcal F}$. For an embedded surface $S$ in $M$ without sphere components, write $\chi(S)$ for its Euler characteristic, $S\cdot S$ for its self-intersection, and $c_1(J_{\mathcal F})\cdot S$ for the evaluation of the first Chern class on $S$. Foliation adjunction conjecture. For every such surface $S$, one has

$$\chi(S)+S\cdot S\leq c_1(J_{\mathcal F})\cdot S.$$

This would generalize P. Kronheimer's theorem for products $N^3\times {\mathbb S}^1$, where the bound is obtained from a taut foliation on the closed irreducible three-manifold $N$. The source presents this as a proposed generalization of that result; its resolution is not specified here.