Principal $2$-Blocks and Sylow $2$-Subgroups

Let $G$ be a finite group with Sylow $2$-subgroup $P⩽G$. 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 $\mathrm{G}\mathrm{a}\mathrm{l}$($\mathbb{Q}$_{|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.