Let $X$ be an compact oriented Riemannian 4-manifold with $b_2^+\geq1$. Let $\omega$ be a self-dual harmonic two form vanishing transversely. On $X\setminus \omega^{-1}(0)$, the spinor bundle $S_+$ can be written as $E\oplus K^{-1}E$, where $E$ is the $-\sqrt{2}i\lvert\omega\rvert$-eigensubbundle of the clifford multiplication $c_+(\omega)$; $K^{-1}E$ is the $+\sqrt{2}i\lvert\omega\rvert$-eigensubbundle.
There is a Spin$^c$ structure such that $E$ is trivial, and a canonical connection $A_0$ on $K^{-1}$ (interpreted as the determinant line bundle) such that the induced spin covariant derivative satisfies that $\nabla_{A_0}(1,0)$ has zero $E$ component.
My question : To establish the following inequality (at a neighbourhood of $\omega^{-1}(0)$) : $$\lvert F_{A_0}\rvert\leq Cdist(\cdot,\omega^{-1}(0))^{-2},$$for some $C$ depending only on the Riemannian metric, $\omega$, and the Spin$^c$ structure.
My attepmt : Since $\omega$ vanishes transversely, I am quite sure the distance function must comes from the inequality $$\lvert\omega\rvert\geq Cdist(\cdot,\omega^{-1}(0)).$$ I tried to develop a differential inequality starting with $d^*d\lvert F_{A_0}\rvert$ and play with the Weitzenbock formula (on forms), however I cannot relate this with $\lvert\omega\rvert$.
On the other hand, I tried to start from $\nabla_{A_0}\nabla_{A_0}\psi$ for some specific spinor $\psi$, however I cannot single out $\lvert F_{A_0}\rvert$ by solely playing with the defining condition on $\nabla_{A_0}(1,0)$.
In fact, $K^{-1}$ can be identify with the anti-canonical bundle of the symplectic manifold $(X\setminus \omega^{-1}(0),\omega)$, however I am not sure whether this is useful or not.
Background : This is an inequality appearing in a paper of Taubes in 1999, on the relationship between Seiberg-Witten theory and nearly symplectic geometry. Taubes stated the inequality without any further explanation. The bound helps proving $\lvert F_{A_0}\rvert$ is integrable over all of $X$ (even crossing $\omega^{-1}(0)$), which is a minor part needed so that the curvature induces a well-defined current over whole $X$.
Any help is appreciated!