Touzet's fibration conjecture for regular foliations on rationally connected manifolds

Let $X$ be a rationally connected projective manifold, and let $\mathcal{F}$ be a regular holomorphic foliation on $X$. A fibration is a morphism

$$X\rightarrow Y$$

onto a projective manifold $Y$.

Touzet's fibration conjecture. The foliation $\mathcal{F}$ is induced by a fibration $X\rightarrow Y$ onto a projective manifold.

This conjecture generalizes the classification of regular one-dimensional holomorphic foliations on smooth projective rational surfaces. It was verified for weak Fano manifolds, but remains open already in dimension $3$.