The nonexistence conjecture for Ulrich Schur bundles on two-step flag varieties

Let $F(k_1,k_2;n)$ be the two-step flag variety parametrizing flags of subspaces of dimensions $k_1$ and $k_2$ in an $n$-dimensional vector space, and let $\mathcal{O}(1)$ denote the polarization used in the paper. A Schur bundle means a homogeneous bundle obtained from a Schur functor on this flag variety. Nonexistence conjecture. The two-step flag variety $F(k_1,k_2;n)$ does not admit a Schur bundle that is Ulrich with respect to $\mathcal{O}(1)$ if $k_1\geq3$ and $k_2-k_1\geq3$. This is the geometric translation of the preceding combinatorial conjecture and concerns the cases not covered by the paper’s partial classifications; its status remains open.