Navarro–Tiep–Vallejo's principal 2-block conjecture

Let $G$ be a finite group and let $P\in\operatorname{Syl}_2(G)$. Let $\sigma\in\operatorname{Gal}(\mathbb{Q}^{\mathrm{ab}}/\mathbb{Q})$ be the unique automorphism that fixes $2$-roots of unity and squares odd roots of unity. Navarro–Tiep–Vallejo's principal 2-block conjecture. The principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if every odd-degree ordinary irreducible character in the principal $2$-block of $G$ is fixed by $\sigma$. This is a block-theoretic analogue of the self-normalizing Sylow 2-subgroup conjecture. The corresponding analogue is known for odd primes, while the case $\ell=2$ is the focus of the paper and remains open.