Equivariant positivity conjecture for CSM classes

Let $G/P$ be a partial flag manifold, let $X(wW_P)^\circ \subseteq G/P$ be a Schubert cell, and write its equivariant CSM class in the Schubert basis as

$${c^{{T}}_{\operatorname{SM}}}(X(wW_P)^\circ) = \sum_{v \le w} c_{v,w}(\alpha) [X(vW_P)]_T \quad \in H_*^T(G/P).$$

Equivariant positivity conjecture. The coefficient $c_{v;w}(\alpha)$ is a polynomial in the positive roots $\alpha$ with non-negative coefficients. Numerical evidence supports this conjecture for equivariant CSM classes in arbitrary partial flag manifolds; the corresponding non-equivariant positivity statement has been proved.