Coadjoint varieties residual-category conjecture

Let $X$ be the coadjoint variety of a simple complex algebraic group $G$. Let $D_s(G)$ denote the short-roots subdiagram of the Dynkin diagram of $G$, which is the entire Dynkin diagram when $G$ is simply laced. Coadjoint varieties residual-category conjecture. The category $\Db(X)$ has an $\operatorname{Aut}(X)$-invariant rectangular Lefschetz exceptional collection with residual category $\cR$ such that: if the Dynkin type of $G$ is $A_n$ with $n$ even, then $\cR=0$; otherwise, $\cR$ is equivalent to the derived category of representations of a quiver of type $D_s(G)$. This conjecture is motivated by known results for homogeneous varieties and gives a uniform prediction for residual categories of coadjoint varieties; its general validity remains open.