Defect-normalizer refinement of the Alperin–McKay–Navarro conjecture

Let $p$ be a prime and $G$ a finite group. Let $B$ be a $p$-block of $G$ with defect group $D$, let $b$ be its Brauer correspondent in $\mathbf{N}_G(D)$, and let $\mathcal{H}_B$ be the subgroup of $\mathcal{H}$ fixing $B$. Defect-normalizer AMN conjecture. There exists an $\mathcal{H}_B$-equivariant bijection

$$^*: \operatorname{Irr}_0(B)\longrightarrow\operatorname{Irr}_0(b)$$

such that, for every $\chi\in\operatorname{Irr}_0(B)$ with $p$-rationality level at least $2$,

$$\mathrm{lev}(\chi_{\mathbf{N}_G(D)})=\mathrm{lev}(\chi^*).$$

This incorporates the defect-normalizer restriction into the Alperin–McKay–Navarro framework and asks the bijection to preserve the relevant $p$-rationality level. The supplied text gives no resolution status.