Equivariant Gal conjecture for crystallographic Coxeter complexes

Let $W$ be a finite crystallographic Coxeter group, and let $\operatorname{Cox}(W)$ be its Coxeter complex. Equivariant Gal conjecture. The pair $(\operatorname{Cox}(W),W)$ exhibits the equivariant Gal phenomenon for every finite crystallographic Coxeter group $W$. This is proposed as an equivariant analogue of the paper's Gal-type gamma-positivity results; the source says it is expected to follow from the classification of finite Coxeter groups, while a proof avoiding classification would be more desirable, and gives no proof or resolution.