The blockwise cyclic-defect conjecture

Let $p\in\{2,3\}$. Let $G$ be a finite group and let $B$ be a $p$-block of $G$ with nontrivial defect group $D$. Let $\sigma_1$ be the Galois automorphism introduced in the paper, fixing $p'$-roots of unity and raising roots of unity of $p$-power order to their $(p+1)$st powers. Write $\operatorname{Irr}_0(B)^{\sigma_1}$ for the height-zero irreducible characters in $B$ fixed by $\sigma_1$. The blockwise cyclic-defect conjecture. One has

$$|\operatorname{Irr}_0(B)^{\sigma_1}|=p\quad\Longleftrightarrow\quad D\text{ is cyclic}.$$

This extends the principal-block criterion proved in the paper to arbitrary blocks and is presented as a consequence expected from the blockwise McKay–Navarro conjecture; it remains open in the source.