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. The complete proof has not been independently verified here. Engel and Mauri (2026) disclose use of ChatGPT Astra for proofreading an existing human-written draft only.
