Non-extendability of complex structures

There exists a complex structure $J$ on a connected open subset $N$ of $S^6$. We prove that: (1) $J$ can be extended to a global almost complex structure $J_1$ on $S^6$; (2) any extension to $S^6$ is necessarily non-integrable. Therefore, it is impossible to deform $J_1$ to an integrable almost complex structure on $S^6$ while fixing it on $N$. This phenomenon indicates that the deformation strategy suggested by S.-T. Yau in his Problem 52 cannot be realized in this sense.