This sequel of Is there a strongly noncommutative fusion category? is motivated to know whether every fusion category is "equivalent" (in some sense) to one with a commutative Grothendieck ring.
A Grothendieck ring will be called strongly noncommutative if every (fusion) categorification $\mathcal{C}$ is strongly noncommutative in the sense of above question (i.e. the Morita equivalent class of $\mathcal{C}$ contains only fusion categories with a noncommutative Grothendieck ring). Above question got an answer providing a strongly noncommutative fusion category which is Grothendieck equivalent to a non strongly-noncommutative one, so it was not completely satisfying for above motivation.
Question: Is there a strongly noncommutative Grothendieck ring?
The Grothendieck rings of two Morita equivalent fusion categories will be called categorically Morita equivalent. This relation may need to be extended by transitivity to be a true equivalence relation (open?), and if so, that leads to the following question:
Bonus question: Is it invariant by categorical Morita equivalence for a Grothendieck ring to be strongly noncommutative?
If not, the definition of strongly noncommutative should be improved accordingly.
For simplicity, all the fusion categories are assumed to be over the complex field.
