The dual stacked upper-bound conjecture for cd-indices of Bruhat intervals

Let $[u,v]$ be an interval in the Bruhat order on a Coxeter group, with $l(u)=k$ and $l(v)=d+k+1$. Let $\Phi_{[u,v]}$ denote its cd-index, and let a dual stacked polytope be a polytope obtained from a simplex by a series of vertex-shavings. The dual stacked upper-bound conjecture. The coefficientwise maximum of all cd-indices $\Phi_{[u,v]}$ with $l(u)=k$ and $l(v)=d+k+1$ is attained on a Bruhat interval which is isomorphic to a dual stacked polytope of dimension $d$ with $d+k+1$ facets. Equivalently, the cd-index of such a dual stacked polytope is coefficientwise an upper bound for the cd-indices of all these Bruhat intervals. The conjecture proposes that Bruhat intervals have no larger cd-index coefficients than the explicitly realized dual stacked examples; the source gives no resolution, so the assertion remains open.