Touzet's conjecture on regular foliations of rationally connected manifolds

Let $X$ be a projective rationally connected manifold and let $\mathcal{F}$ be a regular foliation on $X$. Touzet's conjecture. The foliation $\mathcal{F}$ is induced by a smooth morphism with rationally connected fibers. This conjecture extends the classification of regular rank-one foliations on rational surfaces to higher-dimensional rationally connected manifolds; it is open for $\dim(X) \geq 3$.