Generalized localization equivalence conjecture for Yang–Baxter operators

Let $X$ be a simple object in a fusion category $\mathcal{C}$ equipped with a Yang–Baxter operator $(X,c)$. Let $(YB_{(X,c)},\rho^{(X,c)},\iota)$ denote the associated sequence of Yang–Baxter algebras, braid-group representations, and inclusions. Generalized localization equivalence conjecture. The following conditions are equivalent: the sequence has a (unitary) generalized localization; it has a (unitary) quasi-localization; $\operatorname{FPdim}(X)^2\in\mathbb{N}$; and $\rho^{(X,c)}(\mathcal{B}_n)$ is a finite group for every $n$. This formulation unifies generalized localization, quasi-localization, weak integrality, and finite braid-group image; the source presents it as a version of the earlier localization conjecture, with no resolution stated.