The SYZ conjecture for nef parabolic line bundles on hyperkähler manifolds

Let $X$ be a compact hyperkähler manifold, and let $L$ be a nef line bundle. Let $q$ denote the Beauville–Bogomolov–Fujiki form; $L$ is parabolic when $q(L,L)=0$. SYZ conjecture. Any nef parabolic line bundle on a compact hyperkähler manifold is semiample. The big case is already understood, so the conjecture concerns the isotropic, or parabolic, case and would produce Lagrangian fibrations in the non-trivial projective case.