The nef square-zero divisor criterion for abelian fibrations

Let $X$ be an irreducible holomorphic symplectic manifold, and let $q_X$ denote its Beauville–Bogomolov quadratic form. A divisor is nef if it has nonnegative intersection with every curve, and it is non-trivial if it is not numerically trivial. The nef square-zero criterion. $X$ is an abelian fibration if and only if it contains a non-trivial nef divisor $D$ such that

$$q_X(D)=0.$$

An abelian fibration has such a divisor by pulling back a hyperplane from its base; the converse is expected to follow from suitable vanishing theorems, but is not established in the source.