Berezin-determinant conjecture for invertible objects

Let $\overline{\mathcal{T}}_n$ be the quotient tensor category and let $\mathcal{T}_n$ be the original category. An object $I$ of $\overline{\mathcal{T}}_n$ is invertible if it is an invertible object under the tensor product. Berezin-determinant conjecture. Any invertible object $I$ in $\overline{\mathcal{T}}_n$ is represented in $\mathcal{T}_n$ by a power of the Berezin determinant. This would determine the invertible objects, and hence the character group, of the quotient category.