I was recently introduced to Connes' Embedding Conjecture (CEC) which states:

Every separable type $II_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on $\mathbb{N}$, $R$ is a hyperfinite type $II_{1}$ factor, and $R^{\omega}$ is the ultrapower of $R$ with respect to $\omega$.

I'm not well versed in the theory of factors and this conjecture is mysterious and difficult for me to understand. However, there do seem to be a lot of equivalent formulations of CEC. For some examples of this see: N. Ozawa, "About the Connes Embedding Conjecture---Algebraic Approaches---, arXiv:1212.1700 [math.QA]

My background is in fusion categories, and given the relationship between subfactors and fusion categories I was led to wonder:

**Question:** Are there statements in the theory of fusion categories that are equivalent to CEC, or does CEC have implications for fusion categories?

My understanding of the connections between fusion categories and subfactors, I'm sorry to say, is quite weak. So it very well could be that there is simply no relation between CEC and fusion categories, which would be a completely acceptable answer.