Semisimplicity of the tensor identity in a multifusion category over an arbitary field

For a multifusion category $$\mathcal{C}$$ over an algebraically closed field it is known that $$\text{End}(\mathcal{1})$$ is a commutative semi-simple algebra. See, for example, Theorem 4.3.1 in [1]. In particular this justifies the name 'multifusion category' as $$\mathcal{C}$$ then decomposes into $$\bigoplus_{ij} \mathcal{C}_{ij}$$ where $$i$$ and $$j$$ range over the simple summands of $$\mathcal{1}$$ and $$\mathcal{C}_{ii}$$ is a fusion category.

My question is as follows: under what conditions does this also follow over an arbitrary field? In particular, is it enough to assume that every simple object $$S$$ in $$\mathcal{C}$$ satisfies $$\text{End}(\mathcal{S}) = \mathbb{K}$$ i.e. that the category is split?

[1] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, Tensor categories, Mathematical Surveys and Monographs 205. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2024-6/hbk). xvi, 343 p. (2015). ZBL1365.18001.