Non-equivariant positivity conjecture for CSM classes of Schubert cells

Let $G/B$ be the flag variety of a reductive group $G$ over $\mathbb{C}$, with $B$ a fixed Borel subgroup, and let $W$ be its Weyl group. For $w\in W$, write $Y(w)^\circ=B^-wB/B$ for the opposite Schubert cell and $Y(w)=\overline{Y(w)^\circ}$ for its opposite Schubert variety. Here $c_{\mathrm{SM}}$ denotes the Chern–Schwartz–MacPherson class and $[Y(v)]$ the Schubert-variety class. Non-equivariant positivity conjecture. For $u,v\in W$,

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

where $\mathbb{Z}_{\geq0}$ is the set of nonnegative integers. The conjecture is confirmed in type $A$ for $v$ a Grassmannian permutation of hook shape and has been checked for several small Weyl groups; it remains open in general.