Kumar's positivity conjecture for CSM classes of Richardson cells

Let $G/B$ be the flag variety of a reductive group $G$ over $\mathbb{C}$, with Weyl group $W$. For $u,v\in W$, let $X(u)^\circ$ and $Y(v)^\circ$ be the Schubert and opposite Schubert cells, respectively, and let $c_{\mathrm{SM}}$ denote the Chern–Schwartz–MacPherson class. Kumar's positivity conjecture. For all $u,v\in W$, the CSM class of the Richardson cell $X(u)^\circ\cap Y(v)^\circ$ is effective:

$$c_{\mathrm{SM}}(X(u)^\circ\cap Y(v)^\circ)\in\sum_{w\in W}\mathbb{Z}_{\geq0}\cdot[Y(w)].$$

The paper states that this conjecture is equivalent to the non-equivariant positivity conjecture above; it gives an effective-geometric reformulation, while general resolution is not reported.