The uniruledness and Iitaka-fibration conjecture

Let $X$ be a variety in characteristic $0$. The uniruledness and Iitaka-fibration conjecture. Either $X$ is uniruled, or $\operatorname{Kod}(X)\geq 0$. If $\operatorname{Kod}(X)\geq 0$, then there is an open set in $X$ where the fibers of the Iitaka fibration have Kodaira dimension $0$. This is a conjecture from higher-dimensional classification theory, identified in the source with Conjecture 1.24 of the cited reference. The source states that it is known when the relevant fibers have dimension at most $2$, and consequently the associated construction is unconditional when $\dim X\leq 3$; the conjecture remains open in general.