The minimal-degeneration conjecture for absolutely special vertices

Let $G$ be a reductive group over the relevant field, let $x$ be an absolutely special vertex, and let $\tGr_{G,x}^{\leq \bar{\mu}}$ be the corresponding normalized Schubert variety, with open Schubert cell $\tGr_{G,x}^{\bar{\mu}}$. Minimal-degeneration conjecture. If $x$ is absolutely special, then the smooth locus of $\tGr_{G,x}^{\leq \bar{\mu}}$ is precisely $\tGr_{G,x}^{\bar{\mu}}$. This conjecture extends the boundary-singularity property known for split groups in characteristic zero; the paper highlights it as a conjectural explanation of the behavior for absolutely special vertices, while exotic smoothness shows that the analogous statement can fail for general special vertices.