The fusion-realisation condition for arbitrary fusion rings

Let $R$ be a fusion ring and let $(r_{s,t})_{s,t\in S}$ be a matrix satisfying the geometric conditions for a Coxeter system $(\mathbb W,S)$, as in the fusion-realisation condition: $r_{s,s}=2$, $-r_{s,t}\in R_{\geq 0}$ and $r_{t,s}=r_{s,t}^*$ for $s\ne t$, and the Frobenius–Perron dimensions satisfy the prescribed Coxeter-label conditions. The associated $R$-sesquilinear form on $R\Lambda_S$ defines a geometric faithful $R$-realisation of $(\mathbb W,S)$. Fusion-realisation conjecture. Theorem 6.6 holds for any fusion ring $R$, without assuming that $R$ is categorifiable. The conjecture would remove the extra categorifiability hypothesis from the construction of faithful geometric realisations; whether the theorem extends to all fusion rings is left open in the source.