A fusion category is called simple if its fusion subcategories are just $Vec$ and itself. Let us call a fusion category strongly simple if every fusion category Morita equivalent to it (i.e. same Drinfeld center up to equiv.) is simple.
Question: What are the known examples of strongly simple fusion categories?
Note that if $G$ is a finite group then $Rep(G)$ is simple iff $G$ is simple, but it is strongly simple iff $G$ is cyclic of prime order, because $Rep(G)$ is Morita equivalent to $Vec(G)$ whose fusion subcategories (up to equiv.) are $Vec(H)$ with $H<G$.
The Extended Haagerup fusion categories are strongly simple, see this paper (by Grossman-Morrison-Penneys-Peters-Snyder) classifying their Morita equivalence class. I wrote here the explicit fusion matrices (it is easy to see that they are all simple).
