Conjecture on integrability of split tangent factors on rationally connected manifolds

Let $X$ be a projective manifold with split tangent bundle

$$T_X=V_1 \oplus V_2.$$

A subbundle $V\subset T_X$ is integrable if it is closed under the Lie bracket of local sections. Integrability conjecture. If $X$ is rationally connected, then $V_1$ or $V_2$ is integrable.

There are no known examples of rationally connected manifolds with split tangent bundle whose direct factors are both non-integrable. The conjecture is presented as an optimistic expectation and is proved in the paper only under additional hypotheses.