Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives another fusion category, Morita equivalent to the original $\mathcal{C}$.

Can I bound the rank of $\mathcal{C}_\mathcal{M}^*$?

Recall that the *rank* is the number of isomorphism classes of irreducible objects.
Certainly $\mathcal{C}$ and $\mathcal{C}_\mathcal{M}^*$ have the same global dimension, so easily $\operatorname{rank}(\mathcal{C}_\mathcal{M}^*) \leq \operatorname{dim}(\mathcal{C})$. Are there better upper bounds available?

*Update*: I'm happy to consider all the 'decategorified' data of $\mathcal{C}$ and $\mathcal{M}$, that is, the Grothendieck groups of both, along with the ring and module structures thereon, when trying to come up with an estimate, not just the rank of $\mathcal{C}$.

As examples:

- the Haagerup subfactor gives a Morita equivalence between two fusion categories with ranks 4 and 6, and global dimension $\approx 35.725$
- $\operatorname{Rep}(G)$ and $\text{Vec}_G$ are Morita equivalent, with global dimension $|G|$. Here $\operatorname{rank}(\text{Vec}_G) = |G|$, while when $G$ is non-commutative $\operatorname{rank}(\operatorname{Rep}(G))$ may be much smaller.