Twisted Kazhdan–Lusztig structure-constant unimodality property D-prime

Let $(W,S,*)$ be a triple consisting of a Coxeter system $(W,S)$ and an $S$-preserving involution $*\in\operatorname{Aut}(W)$, with corresponding twisted involutions $\mathbf I_*$. For $x\in W$ and $y,z\in\mathbf I_*$, let $h^+_{x,y;z}$ and $h^-_{x,y;z}$ be the Laurent polynomials defined from the ordinary and twisted structure constants, and let their degrees in $v$ be $d_+$ and $d_-$. Twisted property D-prime. The polynomials $v^{d_+}h^+_{x,y;z}$ and $v^{d_-}h^-_{x,y;z}$ are unimodal polynomials in $q=v^2$. This is presented as an open twisted analogue of the classical unimodality property.