Gamma-positivity conjecture for Coxeter Chow polynomials

Let $W$ be a Coxeter group and let $u<v$ be elements of $W$. The Coxeter Chow polynomial $\mathrm{H}_{uv}$ is the polynomial associated with the Bruhat interval $[u,v]$; it is $\gamma$-positive when it admits a $\gamma$-expansion with nonnegative coefficients. Coxeter Chow gamma-positivity conjecture. Coxeter Chow polynomials of Bruhat intervals of Coxeter groups are always $\gamma$-positive. The conjecture follows from the conjectured nonnegativity of the complete $\mathbf{cd}$-index of Bruhat intervals, while only special cases of that nonnegativity are known.