The sphericality characterization for Schubert varieties

Let $G$ be a complex reductive group with Borel subgroup $B$, Weyl group $W$, and let $X_w=\overline{BwB/B}$ be the Schubert variety associated with $w\in W$. For $I\subseteq J(w)$, let $L_I$ be the corresponding standard Levi subgroup, and call $w$ $I$-spherical when it satisfies the reduced-word conditions defining $I$-sphericity. The sphericality characterization. $X_w$ is $L_I$-spherical if and only if $w$ is $I$-spherical. This conjecture seeks a combinatorial characterization of spherical Schubert varieties under Levi actions; the paper proves it in several special cases, but the general equivalence remains open.