Equality of geometric and representation-theoretic invariants for hyperbolic-like triples

Let $(W,L,\rho)$ be a hyperbolic-like triple, and let ${\mathcal T}_{{\mathcal I}}$ be an ${\mathcal I}$-triangulation of $(W,L,\rho)$. Denote by $c_{{\mathcal I}}(W,L,\rho)$ the corresponding charged, branched ideal triangulation, and let $G(c_{{\mathcal I}}(W,L,\rho))$ and $R(c_{{\mathcal I}}(W,L,\rho))$ be the geometric and representation-theoretic quantities associated with it. Equality conjecture. One has

$$G(c_{{\mathcal I}}(W,L,\rho)) = R(c_{{\mathcal I}}(W,L,\rho)).$$

The conjecture asserts that the geometric and representation-theoretic descriptions agree for hyperbolic-like triples. The stated status evidence indicates that such phenomena have been formally verified in typical situations by several people, so this conjecture is treated as solved.