Inductive Alperin bound conjecture for blocks

Let $G\unlhd A$ be finite groups, let $B$ be a block of $G$ with defect group $D$, and let $b$ be its Brauer correspondent in a subgroup $M$ as below. Write $\operatorname{IBr}(B)$ and $\operatorname{IBr}(b)$ for the sets of irreducible Brauer characters in the respective blocks. Inductive Alperin bound conjecture. There is an $\operatorname{N}_A(D)$-invariant subgroup

$$\operatorname{N}_G(D)\leq M\leq G,$$

with $M<G$ whenever $D$ is not normal in $G$, such that $b$ is the Brauer correspondent of $B$ in $M$ and there exists an $\operatorname{N}_A(D)_B$-invariant injection

$$\Omega:\operatorname{IBr}(b)\hookrightarrow\operatorname{IBr}(B)$$

such that

$$(A_\chi,G,\chi)\succeq_b(M\operatorname{N}_A(D)_\vartheta,M,\vartheta)$$

for every $\vartheta\in\operatorname{IBr}(b)$ and $\chi=\Omega(\vartheta)$. This is a blockwise refinement of Alperin's lower bound and is formulated using block isomorphisms of modular character triples. Its general validity is not established in the supplied text.