Strong positivity conjecture for CSM classes of Schubert cells

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

$${c_{\operatorname{SM}}}(X(wW_P)^\circ) = \sum_{v \le w} c_{v,w} [X(vW_P)] \quad \in H_*(G/P;{\mathbb Z}).$$

Strong positivity conjecture. One has $c_{v,w}>0$ for every $v\le w$. The weaker non-negativity assertion is known, but strict positivity remains the stronger conjectural statement.