Minimality conjecture for the proposed Lefschetz decomposition of Grassmannians

Let $X=\operatorname{Gr}(k,n)$, let $\mathcal U$ be the tautological subbundle, and let $\mathcal A_i$ be the full triangulated subcategories defined in the source using the specified Schur powers of $\mathcal U^*$ and the support function $o(\lambda)$. Minimality conjecture. The categories $\mathcal A_i$ form a minimal Lefschetz decomposition of $\D^b(X)$. If the preceding fullness conjecture holds, this would give a minimal Lefschetz decomposition for the derived category of the Grassmannian; the source gives no resolution of the conjecture.