The degree-monotone McKay bijection conjecture

Let $G$ be a finite group, let $p$ be a prime number, and let $P$ be a Sylow $p$-subgroup of $G$. Write $N_G(P)$ for the normalizer of $P$ in $G$, and let $\operatorname{Irr}_{p'}(G)$ denote the set of irreducible characters of $G$ whose degrees are coprime to $p$. Degree-monotone McKay bijection conjecture. There exists a bijection

$$\varepsilon:\operatorname{Irr}_{p'}(G)\rightarrow\operatorname{Irr}_{p'}(N_G(P))$$

such that $\varepsilon(\chi)(1)\leq\chi(1)$ for all $\chi\in\operatorname{Irr}_{p'}(G)$. This strengthens the McKay conjecture by requiring a bijection that does not increase character degrees; the paper proposes this refinement and proves it for symmetric groups.