Navarro conjecture for finite groups

Let $G$ be a finite group, let $p$ be a prime dividing $|G|$, let $D$ be a Sylow $p$-subgroup of $G$, and let $N_G(D)$ be the normalizer of $D$ in $G$. Let $\operatorname{Irr}(G)$ denote the irreducible characters of $G$, and let $\operatorname{Irr}_{p'}(G)$ and $\operatorname{Irr}_{p'}(N_G(D))$ be the characters of degree relatively prime to $p$ in the respective groups. Let $n=|G|$, let $\mathcal{G}=\operatorname{Gal}(\mathbb{Q}_n/\mathbb{Q})$, and let $\mathcal{N}$ consist of the automorphisms $\sigma_e\in\mathcal{G}$, for nonnegative integers $e$, satisfying $\sigma_e(\xi)=\xi^{p^e}$ for every $p'$-root of unity $\xi$. For $\sigma\in\mathcal{N}$, write $\operatorname{Irr}^{\sigma}_{p'}(G)$ and $\operatorname{Irr}^{\sigma}_{p'}(N_G(D))$ for the subsets fixed by $\sigma$. Navarro conjecture. For every $\sigma\in\mathcal{N}$,

$$|\operatorname{Irr}^{\sigma}_{p'}(G)|=|\operatorname{Irr}^{\sigma}_{p'}(N_G(D))|.$$

This is Navarro's Galois-equivariant refinement of the McKay conjecture. The source calls the underlying McKay conjecture unsolved; it does not provide a resolution of this general Navarro statement, although it proves the relevant assertion for the stated alternating-group setting.