The quasi-line foliation conjecture for projective space

Let $X$ be a projective manifold containing a quasi-line $l$ with $e(X,l)=1$. If the rank-one foliation ${\mathcal F}_x$ satisfies

$$\dim_x {\mathcal F}_x^{\operatorname*{sing}}=0,$$

Quasi-line foliation conjecture. There exists a birational map

$$X\dashrightarrow {\mathbb{P}}^n$$

that is an isomorphism in a neighbourhood of $l$ and maps $l$ onto a line.

This conjecture predicts that the local geometry of a rank-one foliation with an isolated singularity at $x$ forces the ambient manifold to be birational to projective space, compatibly with the given quasi-line. The supplied text presents it as a guess; no resolution is stated here.