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We refer to the book Tensor categories by Etingof-Gelaki-Nikshych-Ostrik (MR3242743) 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).

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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

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