  Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$.
Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$
contains exactly one irreducible Brauer character if and only if all odd-degree
ordinary irreducible characters in the principal $2$-block of $G$ are fixed by
a certain Galois automorphism $\sigma \in
\mathrm{Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$. Recent work of Navarro-Vallejo has
reduced this conjecture to a problem about finite simple groups. We show that
their conjecture holds for all finite simple groups, thus establishing the
conjecture for all finite groups.
