The hyperkähler SYZ conjecture for Lagrangian fibrations

Let $X$ be a hyperkähler manifold. A line bundle $L$ on $X$ is isotropic if its first Chern class satisfies $q_X(c_1(L))=0$, and it is nef if its numerical class has nonnegative intersection with every curve on $X$. A rational Lagrangian fibration means that $X$ is birational to a hyperkähler manifold admitting a Lagrangian fibration.

Hyperkähler SYZ conjecture.

1. $X$ has a Lagrangian fibration if and only if it admits a non-trivial isotropic nef line bundle $L$.
2. $X$ admits a rational Lagrangian fibration if and only if it admits a non-trivial isotropic line bundle $L$.

Nefness is necessary for an algebraic isotropic class to induce a Lagrangian fibration, and this conjecture asserts that it is sufficient. The statement is a hyperkähler reformulation of the SYZ perspective from mirror symmetry and remains open in the generality stated.