Donovan's conjecture for blocks with fixed defect group

Let $\ell$ be a prime, let $k$ be the coefficient field, and let $D$ be a finite $\ell$-group. For a finite group $G$, a block $B$ of $kG$ has defect group isomorphic to $D$ if its defect group is isomorphic to $D$.

Donovan's conjecture. Amongst all finite groups $G$ and blocks $B$ of $kG$ with defect group isomorphic to $D$, there are only finitely many Morita equivalence classes.

If true, this would imply that Morita Frobenius numbers are bounded in terms of a function of the isomorphism class of the defect group. The paper's main theorem shows that Morita Frobenius numbers can nevertheless be arbitrarily large when the defect group varies; the conjecture's status is not resolved in the supplied context.