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 almost complex structure $J$ on a closed ball $B^4$, such that any $C^{\infty}$-small perturbation of $J$ is not tamed by any symplectic form? (I assume that $J$ behaves nicely on the boundary of $B^4$, in particular it is smooth there).

*Remark.* Clearly, if such $J$ exists on $B^4$, it would exists on any $4$-manifold admitting an almost complex structure.

This question is a follow-up to the following one, where a global obstruction for "tamebility" was found for $\mathbb CP^2$ Almost complex structures on $\mathbb CP^2$ that are not tamed