Kuznetsov–Smirnov's residual-category conjecture for coadjoint varieties

Let $X$ be the coadjoint variety of a simple algebraic group $\mathrm G$ over an algebraically closed field of characteristic zero. Let $\mathrm T(\mathrm G)$ be the Dynkin diagram of $\mathrm G$, and let $\mathrm T_{\mathrm{short}}(\mathrm G)$ be its subdiagram formed by the vertices corresponding to short roots. A rectangular Lefschetz exceptional collection on $X$ has residual category $\mathcal R$ as its orthogonal complement. Kuznetsov–Smirnov's conjecture. The category $\mathbf D^{\mathrm b}(X)$ has an $\operatorname{Aut}(X)$-invariant rectangular Lefschetz exceptional collection with residual category $\mathcal R$ such that, if $\mathrm T(\mathrm G)=\mathrm A_n$ and $n$ is even, then $\mathcal R=0$; otherwise, $\mathcal R$ is equivalent to the derived category of representations of a quiver of Dynkin type $\mathrm T_{\mathrm{short}}(\mathrm G)$. The conjecture predicts a uniform description of residual categories for coadjoint varieties; the source gives related constructions and quantum-spectrum evidence, but no general proof or resolution.