SYZ conjecture for potentially Lagrangian classes

Let $Y$ be a hyper-Kähler variety, let $q$ be its Beauville–Bogomolov quadratic form on $H^2(Y)$, and let $l$ be a divisor class on $Y$. The class $l$ is potentially Lagrangian if $q(l)=0$ and $l$ lies on the boundary of the birational Kähler cone; it is Lagrangian if, on a birational hyper-Kähler model $Y'$, it is semi-ample and induces a Lagrangian fibration. SYZ conjecture. Every potentially Lagrangian class is Lagrangian. This conjecture predicts that isotropic classes on the boundary of the birational Kähler cone yield Lagrangian fibrations after birational modification. It is known for all currently known deformation types of hyper-Kähler varieties, but remains open in general.