Donaldson's almost-Kähler a priori bounds conjecture

Let $(M,J, \omega)$ be a compact almost-Kähler manifold, where $J$ is an almost-complex structure and $\omega$ is its fundamental $2$-form. Let $\sigma$ be a smooth volume form on $(M,J, \omega)$ satisfying

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

Suppose that $\overline{ \omega}$ is an almost-Kähler metric on $(M,J)$ such that

$$\overline{ \omega}^n= \sigma,\qquad [\overline{ \omega}]=[ \omega],$$

where the latter means that $\overline{ \omega}- \omega$ is a closed $2$-form. Donaldson's conjecture. There are $C^{\infty}$ a priori bounds on $\overline{ \omega}$ depending only on $\omega$, $J$, and $\sigma$. This conjecture is presented as a curvature-positivity-related motivation; its resolution status is not specified in the source.