I have a surjective smooth map with surjective differential between two balls $\phi:B^{2n}\rightarrow B^{2k}$. Fix an integrable almost complex structure $J$ on $B^{2n}$. Assume that $\mathrm{Ker}\:d\phi$ is preserved by the action of $J$.

For any point $q \in B^{2k}$, I can find a point $p\in B^{2n}$ satisfying $\phi(p)=q$ and I can pushforward the complex structure from $T_{B^{2n}, p}$ to $T_{B^{2k}, q}$. Is it true that the resulting complex structure on $T_{B^{2k}, q}$ does not depend on the lift? If so, do I get an integrable almost complex structure on $B^{2k}$?

I think that the answer to the first question is positive if $\phi^{-1}(q)$ is connected as for sufficiently small open sets in the fiber we have local normal form (and independence from the lift becomes self-evident). However, as Mike Miller mentions, the fibers don't have to be connected.