Rigidity and Flexibility of Schubert classes

Consider a rational homogeneous variety $X$. (For example, take $X$ to be the Grassmannian $\mathrm{Gr}(k,n)$ of $k$-planes in complex $n$-space.) The Schubert classes of $X$ form a free additive basis of the integral homology of $X$. Given a Schubert class $[S]$, represented by a Schubert variety $S$ in $X$, Borel and Haefliger asked: aside from the Schubert variety, does $[S]$ admit any other algebraic representatives? I will discuss this, and related questions, in the case that $X$ is Hermitian symmetric.