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](https://mathoverflow.net/questions/262300/almost-complex-structures-on-mathbb-cp2-that-are-not-tamed)



Almost complex structures on a 4-ball that are not tamed
The following construction provides plenty of examples of non-tamed almost complex structures:


Consider an almost complex structure $J$ on $B^4$ for which the contact hyperplanes of the *overtwisted* contact structure on $S^3=\partial B^4$ (in the same homotopy class as the standard tight contact structure) become $J$-complex. If there was a taming symplectic form, we would have constructed a so-called weak symplectic filling of an overtwisted contact manifold. However, This is not possible by a result of Eliashberg and Gromov.


Finally, observe that the contact condition is open, and by Gray's stability also a perturbation of $J$ would be tangent to $\partial B^4$ along some tangent hyperplane distribution being an *overtwisted* contact structure.


