Nonintegrability conjecture for twists of the standard almost complex structure on the 6-sphere

Let $S^6$ carry its standard almost complex structure $J$, and let $\operatorname{Aut}(TS^6)$ denote the automorphisms of its tangent bundle. For $\psi\in\operatorname{Aut}(TS^6)$, write $J^\psi$ for the twisted almost complex structure associated with $\psi$. Nonintegrability conjecture. The almost complex structure $J^\psi$ is nonintegrable for every $\psi\in\operatorname{Aut}(TS^6)$. This conjecture concerns whether twisting the standard almost complex structure on the 6-sphere can produce an integrable one, a question related to longstanding attempts to understand complex structures on $S^6$.