McKay–Navarro conjecture for principal blocks

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Let $B_0$ be the principal $p$-block of $G$ and $b_0$ the principal $p$-block of $N_G(P)$. Let $\mathcal H$ be the subgroup of Galois automorphisms that act by a power map on roots of unity of order prime to $p$, and let $\operatorname{Irr}_{p'}(B_0)$ denote the irreducible characters in $B_0$ of degree prime to $p$. McKay–Navarro conjecture for principal blocks. Every $\sigma\in\mathcal H$ fixes the same number of characters in $\operatorname{Irr}_{p'}(B_0)$ as in $\operatorname{Irr}_{p'}(b_0)$, namely

$$|\operatorname{Irr}_{p'}(B_0)^\sigma|=|\operatorname{Irr}_{p'}(b_0)^\sigma|.$$

This is the principal-block instance of the blockwise McKay–Navarro conjecture and would imply the preceding 2-generation criterion in the relevant case; the source treats it as open.