Lagrangian fibration conjecture for hyperkähler manifolds

Let $X$ be a hyperkähler manifold of dimension $2r$, and let $L$ be a line bundle on $X$. A Lagrangian fibration is a fibration whose fibers are Lagrangian submanifolds for the holomorphic symplectic form.

Lagrangian fibration conjecture.

(a) If $L$ is a nontrivial nef line bundle on $X$ with $q(L)=0$, then there exists a Lagrangian fibration

$$f:X\longrightarrow \mathbb{P}^r$$

such that

$$L=f^*\mathcal{O}_{\mathbb{P}^r}(k)$$

for some $k\geq 1$.

(b) There exists a hyperkähler manifold $X'$ bimeromorphic to $X$ and a Lagrangian fibration $X'\rightarrow\mathbb{P}^r$ if and only if $X$ admits a line bundle $L\neq\mathcal{O}_S$ with $q(L)=0$.

This is the proposed higher-dimensional analogue of the characterization of elliptic fibrations on K3 surfaces. The source describes it as conjectural and gives no general resolution.