The Lagrangian conjecture for hyperkähler manifolds

Let $X$ be a hyperkähler manifold, and let $L$ be a nontrivial nef line bundle on $X$. Denote by $q(L)$ its Beauville–Bogomolov square. Lagrangian conjecture. If

$$q(L)=0,$$

then some power of $L$ is base-point-free; equivalently, the sections of some power of $L$ define a holomorphic Lagrangian fibration. The conjecture predicts that nef isotropic line bundles on hyperkähler manifolds induce Lagrangian fibrations, extending the connection between isotropic divisor classes and fibrations; the general case remains open, although partial results are known in the meromorphic setting.