Kuznetsov–Smirnov Lefschetz collection conjecture for adjoint varieties

Let $X$ be an adjoint variety that is not quasi-minuscule, let $Y\subset X$ be a general hyperplane section, let $c_1(X)$ and $c_1(Y)$ denote their first Chern indices, and let $\Phi_\aleph$ be the root set used in the paper. A full rectangular Lefschetz collection is a rectangular Lefschetz exceptional collection generating the indicated derived category, and a residual category is the component left after removing the Lefschetz blocks. Kuznetsov–Smirnov conjecture.

1. $D^b(X)$ has a full rectangular Lefschetz collection of size $c_1(X)\times (|\Phi_\aleph|+1)$.
2. $D^b(Y)$ has a rectangular Lefschetz collection of size $c_1(Y)\times (|\Phi_\aleph|+1)$, and its residual category has a completely orthogonal exceptional collection of size $|\Phi_\aleph|+1$. This is a categorical strengthening of the geometric and quantum-cohomological structure of these varieties. The source attributes the expectation to Kuznetsov and Smirnov; no proof of the full statement is supplied there.