Higher-dimensional uniruledness conjecture for fibrations over projective space

Let $X$ be a smooth projective variety, and let $f:X\to\mathbb{P}^n$ be a surjective morphism. Let $\Delta(f)$ denote the discriminant locus of $f$. Higher-dimensional uniruledness conjecture. If either

$$\dim\Delta(f)\le n-2$$

or

$$\deg\Delta(f)\le n+1,$$

then $X$ is uniruled.

This is described as a higher-dimensional analogue of the two-singular-fiber conjecture. The paper proves the case $n=2$ under the additional assumption that the general fiber has a good minimal model, while the parser supplies no resolution status in general.