Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called tamed if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $v$.

Question. Is there an example of an almost complex structure on $\mathbb CP^2$ such that any $C^{\infty}$ small perturbation of $J$ is not tamed?

Added. It turns out that there exists as well a purely local obstruction for any small perturbations of $J$ to be tamed. The precise statement and the answer is here: Almost complex structures on a 4-ball that are not tamed