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Here are two propositions in the book Tensor Categories:

Proposition 9.5.1. A pseudo-unitary fusion category admits a unique spherical structure.

Proposition 9.6.5. Let $\mathcal{C}$ be a weakly integral fusion category defined over $\mathbb{C}$. Then $\mathcal{C}$ is pseudo-unitary.

Question: Is Proposition 9.6.5. true for an integral fusion category defined over a finite field?

The notion of pseudo-unitary (i.e. categorical dimension equals Frobenius-Perron dimension) is defined (in the book) for a fusion category over $\mathbb{C}$, but this notion should exist without problem over a finite field in the integral case.

The combination with Proposition 9.5.1. leads to:

Weaker question: Let $\mathcal{C}$ be an integral fusion category defined over a finite field. Is $\mathcal{C}$ spherical?

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