The transversely projective or surface pull-back conjecture for codimension-one foliations

Let $X$ be a projective manifold, and let $\mathcal{F}$ be a codimension-one foliation on $X$. A singular transversely projective structure means a transversely projective structure allowed to have singularities, and a pull-back by a rational map means the foliation is obtained from a foliation on a complex surface via a rational map.

Conjecture. Every codimension-one foliation on $X$ either admits a singular transversely projective structure, or is the pull-back, by a rational map, of a foliation on a complex surface.

This conjecture extends the classification of pencils and linear spaces of integrable rational one-forms beyond the previously known cases. The source attributes it to several authors and states that it first appeared in the cited literature; no resolution is given here.