The strong exceptional collection conjecture for generalized Grassmannians

Let $\mathbf{G}$ be a simple algebraic group over an algebraically closed field $\mathsf{k}$ of characteristic $0$, and let $\mathbf{P}\subset \mathbf{G}$ be a maximal parabolic subgroup. The strong exceptional collection conjecture. There is a full strong exceptional collection in the bounded derived category of coherent sheaves $D(\mathbf{G}/\mathbf{P})$ consisting of vector bundles. This is the strongest version of the preceding folklore conjecture for generalized Grassmannians. The general assertion remains open, although many classical and sporadic cases are known.