Twisted Kazhdan–Lusztig monotonicity property B-prime

Let $(W,S,*)$ be a triple consisting of a Coxeter system $(W,S)$ and an $S$-preserving involution $*\in\operatorname{Aut}(W)$, and let $\mathbf I_*$ be the corresponding set of twisted involutions. For $y,w\in\mathbf I_*$ define

$$P^\pm_{y,w}=\frac{1}{2}(P_{y,w}\pm P^\sigma_{y,w}).$$

Twisted property B-prime. The polynomials $P^\pm_{y,w}$ are decreasing for fixed $w$: whenever $y,z,w\in\mathbf I_*$ and $y\leq z$, both differences $P^+_{y,w}-P^+_{z,w}$ and $P^-_{y,w}-P^-_{z,w}$ have nonnegative coefficients. This is presented as part of the conjectural twisted positivity package.