We refer to the book *Tensor categories* by Etingof-Gelaki-Nikshych-Ostrik ([MR3242743](https://mathscinet.ams.org/mathscinet-getitem?mr=3242743)) for the notion of (unitary) fusion category. Two fusion categories are *Grothendieck equivalent* if they have the same fusion ring. 


**Question**: Is there a fusion category not Grothendieck equivalent to a unitary one?


The following citations (coming from above book) are almost on-topic, but not exactly, because they are about the existence of a certain structure on certain fusion categories, whereas above question relaxes up to Grothendieck equivalence, which is much weaker.


On page 284: 



> 
> We note that we do not know an example of a fusion category over
>  $\mathbb{C}$ which does not admit a Hermitian structure, or a
>  pseudo-unitary fusion category which does not admit a unitary
>  structure.
> 
> 
> 


On page 76: 



> 
> Does every semisimple tensor category admit a pivotal structure? A
>  spherical structure? This is the case for all known examples. The
>  general answer is unknown to us at the moment of writing (even for
>  fusion categories over ground fields of characteristic zero).
> 
> 
> 



Is there a fusion category not Grothendieck equivalent to a unitary one?
Yes, according to Andrew Schopieray. He just provided a categorifiable fusion ring, of rank 6 and multiplicity 2, without pseudounitary categorification (so without unitary categorification), in the following preprint called *Non-pseudounitary fusion*.  

<https://arxiv.org/abs/2010.02958>


