Sign conjecture for the complete cd-index of Bruhat intervals

Let $(W,S)$ be a Coxeter system and let $u,v\in W$ with $u<v$. For a ${\bf cd}$-word $w$, write $[w]_{u,v}$ for its coefficient in the complete ${\bf cd}$-index of the Bruhat interval $[u,v]$.

Sign conjecture. For every ${\bf cd}$-word $w$, one has

$$[w]_{u,v}\ge 0.$$

Every Bruhat interval is shellable and Eulerian, hence Gorenstein$^{*}$. By work of Karu, the asserted nonnegativity is already known when $\deg w=l(u,v)-1$, the top degree for which $[w]_{u,v}\ne0$; the conjecture remains open in lower degrees.