The existence conjecture for full exceptional collections on projective homogeneous varieties

Let $X$ be a projective homogeneous variety of a split semisimple linear algebraic group $G$ over a field of characteristic zero.

Exceptional collection conjecture. There exists a full exceptional collection of vector bundles in $\mathsf{D}^{\mathrm{b}}(X)$.

The conjecture proposes that projective homogeneity forces the bounded derived category of coherent sheaves on $X$ to admit a full exceptional collection of vector bundles. The paper constructs such collections in several rank-two cases and rules out collections of the expected length for type $G_2$; its general status is not established here.