Twisted Kazhdan–Lusztig positivity package

Let $(W,S,*)$ be a triple in which $(W,S)$ is a Coxeter system and $*\in\operatorname{Aut}(W)$ is an $S$-preserving involution. Let Properties A-prime, B-prime, C-prime, and D-prime be the four twisted positivity and unimodality properties for the polynomials $P^\pm_{y,w}$ and $h^\pm_{x,y;z}$ defined above. Twisted positivity package. Properties A$'$, B$'$, C$'$, and D$'$ hold for all such triples $(W,S,*)$. The paper's main purpose is to provide evidence for this conjecture, rather than to establish it in full.