Generic vanishing conjecture for the J-anti-invariant cohomology

Let $M$ be a compact 4-manifold and let $J$ be an almost complex structure on $M$. Denote by $H_J^-$ the $J$-anti-invariant part of the second cohomology and set $h_J^- = \dim H_J^-$. Generic vanishing conjecture. For generic almost complex structures $J$ on a compact 4-manifold $M$,

$$h_J^- = 0.$$

This predicts that the $J$-anti-invariant cohomology generically vanishes, complementing the relation between the dimensions of the $J$-invariant and $J$-anti-invariant cohomology and the second Betti number. The source provides no resolution, so the conjecture remains open.