Uniqueness conjecture for fully faithful representations of unitary fusion categories in bimodules

Let $\mathcal{C}$ be a unitary fusion category, and let $R$ be a hyperfinite factor which is either of type ${\rm II}_1$ or ${\rm III}_1$. A fully faithful representation of $\mathcal{C}$ in $\operatorname{Bim}(R)$ is a fully faithful functor $\mathcal{C}\to\operatorname{Bim}(R)$; two such representations are equivalent in the sense of Definition. Uniqueness conjecture. Any two fully faithful representations

$$\mathcal{C}\to\operatorname{Bim}(R)$$

are equivalent. The conjecture is motivated by Popa's uniqueness theorems for hyperfinite finite-depth subfactors of types ${\rm II}_1$ and ${\rm III}_1$, but the source does not prove it.