Non-integrable nonnegative-Ricci metrics on high blowups of the projective plane

Determine whether the four-manifolds \(\mathbb{CP}^2\#k\overline{\mathbb{CP}}{}^{2}\) for \(k\geq 9\) admit an almost Kähler metric with nonnegative Ricci curvature and a non-integrable compatible almost complex structure.

Background

The paper classifies closed almost Kähler four-manifolds with nonnegative Ricci curvature according to their underlying diffeomorphism type and the integrability of the compatible almost complex structure. For rational surfaces with at most eight blowups of $\mathbb{CP}^2$, a perturbation of a Kähler metric with positive Ricci curvature supplies examples in which the almost complex structure is non-integrable.

For $k\geq 9$, the authors state that it is unknown whether such examples exist. The perturbation argument used for del Pezzo surfaces does not apply because these higher blowups do not admit the required starting Kähler metric with positive Ricci curvature.