The strong Frobenius number conjecture for blocks

Let $p$ be a prime, let $(K,\mathcal{O},k)$ be a $p$-modular system, and let $B$ be a block of $\mathcal{O}G$ for a finite group $G$. The strong $\mathcal{O}$-Frobenius number $\operatorname{sf}_{\mathcal{O}}(B)$ is the smallest integer $n$ for which there is an $\mathcal{O}$-algebra isomorphism $B\to B^{(p^n)}$ whose induced $\overline K$-algebra isomorphism sends every $\chi\in\operatorname{Irr}(B)$ to $\chi^{(p^n)}$. Strong Frobenius number conjecture. Let $P$ be a finite $p$-group. There is $s(P)\in\mathbb N$ such that, if $G$ is a finite group and $B$ is a block of $\mathcal{O}G$ with defect groups isomorphic to $P$, then $\operatorname{sf}_{\mathcal{O}}(B)\leq s(P)$. Equivalently, there is $t(P)\in\mathbb N$ such that the $\mathcal{O}$-Morita Frobenius number of every such block is at most $t(P)$. Bounds on these Frobenius numbers are used to reduce Donovan's conjecture to bounding Cartan invariants of blocks of quasisimple groups. The conjecture is not resolved in general.