Cerveau–Lins Neto conjecture on foliations with infinite transverse action

Let $X$ be a projective manifold and let $\mathcal F$ be a codimension one foliation on $X$. An endomorphism $f\in\operatorname{End}(X,\mathcal F)$ is one preserving $\mathcal F$, and its transverse action is the induced action on the transverse structure of $\mathcal F$. The foliation is virtually transversely additive when, after passing to a suitable finite cover, its transverse structure is additive.

Cerveau–Lins Neto conjecture. If the transverse action of $f\in\operatorname{End}(X,\mathcal F)$ is infinite, then $\mathcal F$ is virtually transversely additive.

This conjecture predicts that the transverse-projective and purely transcendental hypothesis used in the preceding structure theorem is unnecessary. The supplied text gives no resolution of the conjecture.