  In our previous paper we suggested a conjecture relating the structure of the
small quantum cohomology ring of a smooth Fano variety to the structure of its
derived category of coherent sheaves. Here we generalize this conjecture, make
it more precise, and support by the examples of (co)adjoint homogeneous
varieties of simple algebraic groups of Dynkin types $A_n$ and $D_n$, i.e.,
flag varieties $Fl(1,n;n+1)$ and isotropic orthogonal Grassmannians $OG(2,2n)$;
in particular we construct on each of those an exceptional collection invariant
with respect to the entire automorphism group. For $OG(2,2n)$ this is the first
exceptional collection proved to be full.
