Donaldson's Ricci-form conjecture for almost-Hermitian four-manifolds

Let $(M,\Omega)$ be a compact symplectic four-manifold with $b^+(M)=1$, equipped with an almost complex structure $J$ tamed by $\Omega$. Let $g_{\Omega}$ be the metric determined by $(\Omega,J)$, and let $\operatorname{Ric}(g_{\Omega},J)$ denote the canonical-connection Ricci form. For a compatible symplectic form $\tilde{\omega}$, write $\tilde{g}$ for its associated metric. Donaldson's Ricci-form conjecture. For every smooth function $F$ on $M$, there exists a unique symplectic form $\tilde{\omega}\in[\Omega]$ compatible with $J$ satisfying

$$\operatorname{Ric}(\tilde{g},J)=\operatorname{Ric}(g_{\Omega},J)+\frac{1}{2}d(JdF).$$

This is proposed as the analogue of Yau's prescribed-Ricci-form theorem in the almost-complex setting. Its status is left open in the source.