Fullness conjecture for the minimal 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 generated by the specified Schur powers of $\mathcal U^*$ with $i<o(\lambda)$. The categories $\mathcal A_i(i)$ are already known to be semi-orthogonal and generate a full triangulated subcategory of $\D^b(X)$. Fullness conjecture. The categories $\mathcal A_i(i)$ generate $\D^b(X)$; equivalently, the categories $\mathcal A_i$ form a Lefschetz decomposition of $\D^b(X)$. This would establish fullness of the proposed Lefschetz decomposition, complementing the paper's theorem proving semi-orthogonality; the source gives no resolution of the conjecture.