Bounded Morita Frobenius numbers for fixed defect groups

Let $\ell$ be a prime, let $k$ be the coefficient field, and let $D$ be a finite $\ell$-group. For a finite group $G$ and a block $B$ of $kG$, write $\operatorname{mf}(B)$ for its Morita Frobenius number.

Bounded Morita Frobenius number conjecture. Amongst all finite groups $G$ and blocks $B$ of $kG$ with defect group isomorphic to $D$, the Morita Frobenius number $\operatorname{mf}(B)$ is bounded.

This is identified in the source with Question 6.1 of the cited work and is presented as a consequence of interest related to Donovan's conjecture. The paper proves that Morita Frobenius numbers are arbitrarily large when the defect group varies, but does not resolve this fixed-defect-group boundedness statement.