Non-negativity conjecture for cd-indices of Bruhat intervals

Let $W$ be a Coxeter group, let $u,v\in W$ satisfy $u\leq v$ in the Bruhat order, and let $\Phi_{[u,v]}$ denote the cd-index of the interval $[u,v]$. Bruhat-interval non-negativity conjecture. For any $u\leq v$ in $W$, the coefficients of $\Phi_{[u,v]}$ are non-negative. This is presented as the conjectured lower bound for cd-indices of Bruhat intervals and as a special case of Stanley's conjecture. The source gives no resolution, so the assertion remains open here.