Positivity of Chern numbers for irreducible hyper-Kähler manifolds

The known irreducible hyper-Kähler manifolds include Hilbert schemes of points on K3 surfaces, generalized Kummer varieties, and two further examples in dimensions $6$ and $10$. A monomial Chern number is a Chern number obtained by integrating a monomial in the Chern classes over the manifold. Chern-number positivity conjecture. All monomial Chern numbers of irreducible hyper-Kähler manifolds are positive. This conjecture is motivated by positivity observed in all known examples and was asked previously by Nieper-Wißkirchen and raised again by Oberdieck, Shen, and Vial; its general validity remains open.