Kronheimer's adjunction inequality conjecture for taut foliations on 4-manifolds

Let $M^4$ be a $4$-manifold, let ${\mathcal F}$ be a taut foliation on $M$, and let $J_{\mathcal F}$ be an almost-complex structure induced by ${\mathcal F}$. For an embedded surface $S$ 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 pairing with the first Chern class. Kronheimer's conjecture. For every such surface $S$, one has

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

This would generalize Kronheimer's theorem for product foliations on $N^3\times {\mathbb S}^1$, where $N$ is a closed irreducible $3$-manifold. The analogous statement for general taut foliations on $4$-manifolds is presented as a possible generalization and is not established in the source.