Kollár's divisor criterion for genus one fibrations of Calabi–Yau manifolds

Let $X$ be a Calabi–Yau $n$-fold. A divisor $D\in H^{2}(X,\mathbb{Q})$ satisfies the numerical conditions

$$D^n=0,\qquad D^{n-1}\neq 0,$$

and is nonnegative on every algebraic curve $C\subset X$ if $D\cdot C\geq 0$.

Kollár's conjecture. $X$ is genus one (or elliptically) fibered if and only if there exists a divisor $D\in H^{2}(X,\mathbb{Q})$ satisfying

$$D^n=0,\qquad D^{n-1}\neq 0,$$

and $D\cdot C\geq 0$ for all algebraic curves $C\subset X$.

This criterion characterizes genus one or elliptic fibrations using the intersection form and positivity against algebraic curves. The source presents it as a conjecture attributed to Kollár; no resolution status is supplied here.