Sign conjecture for coefficients of orbit-stratification characteristic classes

Let $S_n$ be the symmetric group, let $p,w\in S_n$, and let $\mathcal{C}[p]$ be the characteristic class associated with $p$, expanded in the basis $\\{[w]:w\in S_n\\}$.

Sign conjecture. The coefficient of $[w]$ in the expansion of $\mathcal{C}[p]$ is a Laurent polynomial in $\tau_1,\ldots,\tau_n,y$ whose terms have sign $(-1)^{\ell(p)-\ell(w)}$.

The conjecture is supported by direct verification in the displayed formulas and by verification for larger flag varieties with $n\leq 5$; its general case remains open.