The unfolding theorem for arbitrary fusion rings

Let $(\mathbb W,S)$ be a Coxeter system with a geometric $R$-realisation over a fusion ring $R$, and let $(\check{\mathbb W},\check{S})$ be the corresponding unfolded Coxeter system. Let $\pi:\check{S}\to S$ be the partition sending $(b,s)$ to $s$, and consider the associated homomorphism of fundamental groups from the embedding of regular orbit spaces. Unfolding conjecture. Theorem 5.2.7 holds for all fusion rings $R$: the partition $\pi$ is a strong admissible partition. In particular, the homomorphism of fundamental groups in Theorem 6.4 is always a strong admissible homomorphism associated to a strong admissible folding. This would show that the folding and hyperplane-complement construction does not require categorifiability of the fusion ring; the claim remains open in the source.