Navarro's self-normalizing Sylow 2-subgroup 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's self-normalizing Sylow 2-subgroup conjecture.

$$N_G(P)=P$$

if and only if all odd-degree irreducible characters of $G$ are fixed by $\sigma$. The conjecture was subsequently established through reductions to finite simple groups and results proving that every finite simple group has the required property.