Kuznetsov–Smirnov 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 $\operatorname{Aut}(X)$ denote its automorphism group, let $\mathcal{R}$ be the residual category, let $\mathrm{T}(\mathrm{G})$ be the Dynkin type of $\mathrm{G}$, and let $\mathrm{T}_{\mathrm{short}}(\mathrm{G})$ be the indicated short Dynkin type. Kuznetsov–Smirnov 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 $\mathrm{T}_{\mathrm{short}}(\mathrm{G})$. This is a refinement relating the structure of the derived category to the quantum cohomology of coadjoint varieties. The source reports that it has been proved in all cases except types $\mathrm{E}_6$, $\mathrm{E}_7$, and $\mathrm{E}_8$.