A priori bounds for the almost-complex Calabi–Yau equation

Let $X$ be a compact $4$-manifold, let $\Omega$ be a symplectic form on $X$, and let $\mathcal{P}$ be a constraint manifold defined by an almost-complex structure tamed by $\Omega$. Fix a smooth volume form. A priori-bounds conjecture. There are $C^{\infty}$ a priori bounds on every closed form $\omega\subset\mathcal{P}$ satisfying

$$[\omega]=[\Omega].$$

This conjecture predicts that taming by a symplectic form prevents blow-up of solutions in the prescribed cohomology class, extending the corresponding Calabi–Yau picture from integrable complex structures to almost-complex structures. The supplied text does not indicate whether the conjecture has been resolved.