Weak abundance conjecture for uniruled varieties

Let $X$ be a projective variety. A variety is uniruled if it is covered by rational curves, and its Kodaira dimension is denoted by $\kappa(X)$.

Weak abundance conjecture. $X$ is uniruled if and only if it has negative Kodaira dimension:

$$X\text{ is uniruled}\quad\Longleftrightarrow\quad \kappa(X)<0.$$

The conjecture is known in dimension at most three; its general case remains open.