Chern classes of Schubert cells and varieties

Chern-Schwartz-MacPherson class is a functorial Chern class defined for any algebraic variety. I will give a geometric proof of a positivity conjecture of Aluffi and Mihalcea that Chern classes of Schubert cells and varieties in Grassmannians are positive. While the positivity conjecture is a purely combinatorial statement, a combinatorial 'counting' proof is known only in very special cases. In addition, the current geometric argument do not work for Schubert varieties in more general flag varieties. Does the same positivity for Chern classes of Schubert varieties in $G/P$?