The CM-field and Hermitianity conjecture for fusion categories

Let $\mathcal{C}$ be a modular, ribbon or braided fusion category over $\mathbb{C}$. A category is Hermitian over a field $K$ when it admits the corresponding Hermitian structure defined over $K$. CM-field and Hermitianity conjecture. The category $\mathcal{C}$ can be defined over some CM number field $K$ and is Hermitian over $K$. This would extend the preceding arithmetic and Hermitianity results from the associated quantum representations to the categories themselves. The source does not state a resolution.