Donald's a priori estimate conjecture for the symplectic Calabi–Yau equation

Let $(M^4,J)$ be a closed almost-complex $4$-manifold, let $\omega$ be a symplectic form taming $J$, let $\tilde{\omega}$ be a cohomologous symplectic form compatible with $J$, and let $\sigma$ be a smooth positive volume form satisfying

$$\int_M\sigma=\int_M\omega^2.$$

The Calabi–Yau equation is

$$\tilde{\omega}^2=\sigma.$$

Donald's conjecture. For any $k\geqslant 0$, one can bound $\|\tilde{\omega}\|_{C^k(M,g)}$ by a constant depending only on $k$ and on bounds for $\sigma$, $\omega$, and $(M,J)$, where $g$ is the Hermitian metric associated with $\omega$ and $J$. This is Donaldson's proposed symplectic generalization of the Calabi conjecture; the paper proves an a priori estimate in this setting, while the full conjectural statement remains open.