The isotropic line bundle conjecture for Lagrangian fibrations on holomorphic symplectic manifolds

Let $X$ be an irreducible holomorphic symplectic manifold of dimension $2g$, and let $q_X$ denote its Beauville–Bogomolov form. A line bundle $L$ on $X$ is called non-trivial if it is not the trivial line bundle. The isotropic line bundle conjecture. If there exists a non-trivial line bundle $L$ on $X$ satisfying

$$q_X(L)=0,$$

then $X$ admits a Lagrangian fibration, at least up to birational equivalence. This predicts that an isotropic line bundle is the numerical source of a Lagrangian fibration on an irreducible holomorphic symplectic manifold; the statement is presented in the paper as a question/conjecture and its resolution is not established in the supplied context.