Integrability conjecture from J-anti-invariant cohomology

Let $M$ be a compact 4-manifold, let $J$ be an almost complex structure on $M$, and let $H_J^-$ denote the $J$-anti-invariant part of the second cohomology, with $h_J^- = \dim H_J^-$. Integrability conjecture. If

$$h_J^- \geq 3,$$

then $J$ is integrable. This conjecture proposes a numerical criterion for integrability of an almost complex structure on a compact 4-manifold. The source provides no resolution, so the conjecture remains open.