The birational parametrization conjecture for highly movable rational curves

Let $W$ be an arbitrary projective variety of dimension $n$. Assume that $W$ contains a rational curve and that every rational curve of $W$ moves in a family of dimension at least $2n-2$. Birational parametrization conjecture. There is a birational dominating morphism

$${\Bbb P}^n \longrightarrow W$$

that is either an isomorphism or a normalization. The smooth case is supplied by Miyaoka's theorem, while the extension to arbitrary projective varieties is stated as an expectation and remains open.