A fusion category is called *noncommutative* if its Grothendieck ring is noncommutative. Let us call a fusion category *strongly noncommutative* if every fusion category Morita equivalent to it (i.e. same Drinfeld center up to equiv.) is noncommutative.

**Question**: Is there a strongly noncommutative fusion category (say over $\mathbb{C}$)?

If so, what are the known examples?

Note that if $G$ is a finite group then $Vec(G)$ is not strongly noncommutative (even if $G$ is noncommutative) because it is Morita equivalent to $Rep(G)$ which has a commutative Grothendieck ring. Moreover, the Extended Haagerup fusion categories are also not strongly noncommutative because they form a Morita equivalent class and one of them is commutative.

This post is in the same spirit than this one about strongly simple fusion categories. The main difference is that I know examples of strongly simple fusion categories whereas I do not know a single strongly noncommutative fusion category.

The next step would be about fusion categories which are both strongly simple and strongly noncommutative.

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