The two-singular-fiber uniruledness conjecture

Let $X$ be a smooth projective variety, and let $f:X\to\mathbb{P}^1$ be a surjective morphism with at most $2$ singular fibers. Two-singular-fiber uniruledness conjecture. Then $X$ is uniruled.

The claim is presented as following from the non-vanishing conjecture together with a result of Viehweg–Zuo; the parser supplies no resolution status for it.