The 2-generated Sylow 2-subgroup conjecture

Let $G$ be a finite group and let $P$ be a Sylow $2$-subgroup of $G$, with $4\mid |G|$. Let $B_0(G)$ be the principal $2$-block, and let $\operatorname{Irr}_{2'}(B_0(G))^{\sigma_1}$ denote the irreducible characters in that block of degree prime to $2$ fixed by $\sigma_1$. The 2-generated Sylow 2-subgroup conjecture. One has

$$|\operatorname{Irr}_{2'}(B_0(G))^{\sigma_1}|=4\quad\Longleftrightarrow\quad P\text{ is 2-generated}.$$

The paper notes that this follows whenever the relevant principal-block case of the blockwise McKay–Navarro conjecture holds; it proves the assertion when $P$ is normal, but leaves the general statement as a conjecture.