Reading's maximum cd-index conjecture for Bruhat intervals

Let $(W,S)$ be a Coxeter system, and let $[u,v]$ be an interval in the Bruhat order of $W$. Write $k$ for the length of $u$ and suppose that the length of $v$ is $n+k+1$. The $cd$-index of a dual stacked $n$-polytope with $n+k+1$ facets is understood coefficientwise. Reading's conjecture. The $cd$-index of $[u,v]$ is coefficientwise less than or equal to the $cd$-index of a dual stacked $n$-polytope with $n+k+1$ facets. In particular, the $cd$-index of an interval $[1,v]$ is less than or equal to the $cd$-index of a Boolean lattice. Bruhat intervals are Eulerian and shellable, and Reading showed that their $cd$-indices span the generalized Dehn--Sommerville space. The proposed coefficientwise maximum remains unresolved in the source.