The exceptional collection conjecture for semisimple homogeneous varieties

Let $G$ be a semisimple algebraic group over an algebraically closed field of characteristic $0$, and let $P\subset G$ be a parabolic subgroup. The homogeneous variety $G/P$ is the quotient of $G$ by $P$.

Exceptional collection conjecture. There is a full exceptional collection of vector bundles on $G/P$.

This conjecture extends the known constructions of full exceptional collections on projective spaces, Grassmannians, flag varieties, and smooth quadrics. The source does not specify a resolution of the conjecture, so its status is open.