On the derived category of the adjoint Grassmannian of type F

We construct a full rectangular Lefschetz collection in the derived category of the adjoint Grassmannian in type $\mathrm{F}_4$. This gives the first example of a full exceptional collection on this variety and also proves a conjecture due to Alexander Kuznetsov and the author that relates the structure of the derived category of coherent sheaves to the small quantum cohomology in the case of adjoint non-coadjoint varieties.