# Is an integral fusion category pseudo-unitary over a finite field?

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?