Minimal-dimensional modular extension conjecture for finite braided tensor categories

Let $\2C$ be a braided tensor $C^*$-category of finite dimension, and let $\2Z(\2C)$ denote its center. A modular extension of $\2C$ is a modular category $\2M$ containing $\2C$ as a full subcategory. Minimal-dimensional modular extension conjecture. For every braided tensor $C^*$-category $\2C$ of finite dimension there exists a modular extension $\2M$ satisfying

$$\dim\2M=\dim\2C\cdot\dim\2Z(\2C).$$

The preceding lemma proves the corresponding lower bound for any modular extension. Thus the conjecture asserts that this bound is always attained, but the source supplies no proof or resolution.