Alternating-sign expansion conjecture for CSM classes of Richardson cells

Let $G$ be a connected simple algebraic group over $\mathbb{C}$, let $B$ be a Borel subgroup, and let $W$ be its Weyl group with length function $\ell$. For $u,v\in W$, let $\mathring{X}^v_u$ be the Richardson cell, and write its CSM class in the CSM basis of Schubert cells as

$$c_{\operatorname{SM}}(\mathring{X}^v_u)=\sum_{w\in W}d^{u,v}_w c_{\operatorname{SM}}(\mathring{X}_w).$$

Alternating-sign conjecture. For every $u,v,w\in W$,

$$(-1)^{\ell(w)-\ell(u)-\ell(v)}d^{u,v}_w\geq 0.$$

This is a weaker form of the coefficientwise positivity conjecture in the Schubert basis, and the paper proves that the stronger conjecture implies it. The general validity of this weaker sign condition remains open.