Conjecture on residual categories of coadjoint varieties

Let $X$ be the coadjoint variety of a simple algebraic group $\mathrm{G}$ over an algebraically closed field of characteristic zero. Coadjoint-variety 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:

1. If $\mathrm{T}(\mathrm{G})=\mathrm{A}_n$ and $n$ is even, then $\mathcal R=0$.
2. Otherwise, $\mathcal R$ is equivalent to the derived category of representations of a quiver of Dynkin type $\mathrm{T}_{\mathrm{short}}(\mathrm{G})$.

This conjecture predicts the residual category from the short-root subdiagram of the Dynkin diagram. It is presented as a consequence of the quantum-spectrum calculation together with the main conjecture, but the source gives no proof or resolution status.