The strong hyper-Kähler SYZ conjecture asserts that a holomorphic line bundle 
on a compact irreducible
holomorphic symplectic manifold 
is semiample when its first Chern
class is nonzero, lies in the closure of the Kähler cone, and is isotropic
for the Beauville-Bogomolov-Fujiki
form. Semiampleness means that some positive tensor power is generated by global
holomorphic sections on the given manifold.
Engel and Mauri (2026) reported a proof that every nef line bundle on a compact irreducible holomorphic symplectic manifold is semiample. In particular, a nontrivial isotropic nef line bundle induces a Lagrangian fibration. Engel and Mauri (2026) also proved finiteness of deformation classes in any fixed complex dimension when the second Betti number is at least 5.
