The conjecture on full exceptional collections for projective homogeneous spaces

Let $G$ be a semisimple algebraic group and let $G/P$ be a projective homogeneous space of $G$. A collection of vector bundles on $G/P$ is exceptional if its derived endomorphism condition gives the base field for each object and vanishing derived morphisms in the prescribed order; it is full if it generates the derived category of coherent sheaves on $G/P$. Homogeneous-space exceptional-collection conjecture. Any projective homogeneous space of a semisimple algebraic group admits a full exceptional collection consisting of vector bundles. Full exceptional collections would give an explicit description of the derived category of coherent sheaves on projective homogeneous spaces. The statement is presented as a conjecture in the source, and no resolution is supplied here.