Billera–Brenti's strong conjecture on complete cd-indices of Bruhat intervals

Let $W$ be a Coxeter group with identity element $e$, let $w\in W$, and let $[e,w]$ be the corresponding Bruhat interval. Its complete cd-index is the polynomial $\widetilde{\Phi}_{ew}(\texttt{c},\texttt{d})$, while $\Phi_{B_{\ell(w)}}(\texttt{c},\texttt{d})$ denotes the cd-index of the Boolean lattice of rank $\ell(w)$, where $\ell(w)$ is the Coxeter length of $w$.

Billera–Brenti's strong conjecture. One has the coefficientwise inequality

$$\widetilde{\Phi}_{ew}(\texttt{c},\texttt{d})\leq \Phi_{B_{\ell(w)}}(\texttt{c},\texttt{d}).$$

This strengthens the corresponding conjecture for the ordinary cd-index, attributed here to Reading. The claim concerns the positivity and coefficientwise boundedness of complete cd-indices of Bruhat intervals by the Boolean-lattice cd-index; its resolution is not specified in the source.