Sylow-AWC conjecture

Fix a prime $p$. Let $G\unlhd A$ be finite groups, let $P$ be a Sylow $p$-subgroup of $G$, and write $\operatorname{IBr}(H)$ for the set of $p$-Brauer characters of a group $H$. Sylow-AWC conjecture. There is an $\operatorname{N}_A(P)$-invariant subgroup

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

with $M<G$ whenever $P$ is not normal in $G$, and an $\operatorname{N}_A(P)$-invariant injection

$$\Omega:\operatorname{IBr}(\operatorname{N}_G(P))\hookrightarrow\operatorname{IBr}(G)$$

such that

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

for every $\vartheta\in\operatorname{IBr}(\operatorname{N}_G(P))$ and $\chi=\Omega(\psi)$. This is the block-free version of the inductive Alperin-bound condition and is used to reduce the corresponding problem to finite simple groups. The supplied text does not establish the conjecture in general.