Full exceptional collections of homogeneous bundles on rational homogeneous varieties

Prove that the bounded derived category Db(G/P) of any rational homogeneous variety G/P admits a full exceptional collection consisting of G-homogeneous vector bundles, a conjecture that remains open even for irreducible Hermitian symmetric spaces (with the isotropic Lagrangian Grassmannians LG(n) recently settled).

Background

Kapranov constructed full exceptional collections for projective spaces, quadrics, and Grassmannians, using homogeneous bundles. Motivated by these results, a folklore conjecture posits that any rational homogeneous variety G/P has a full exceptional collection of G-homogeneous bundles.

The authors note that while recent work has established fullness for LG(n), the conjecture remains open more broadly, including in the class of irreducible Hermitian symmetric spaces.