Divisibility refinement for p-solvable groups

Let $p$ be a prime, let $G$ be a $p$-solvable finite group, and let $P$ be a Sylow $p$-subgroup of $G$. Let $\operatorname{N}_{G}(P)$ be the normalizer of $P$, and let $\operatorname{Irr}_{p'}(G)$ denote the irreducible characters of $G$ whose degrees are not divisible by $p$. Divisibility refinement. There exists a bijection

$$f:\operatorname{Irr}_{p'}(G)\longrightarrow \operatorname{Irr}_{p'}(\operatorname{N}_{G}(P))$$

such that

$$f(\chi)(1)\mid\chi(1)\quad\text{for all }\chi\in\operatorname{Irr}(G).$$

This strengthens the numerical McKay assertion by requiring a degree-divisibility property for the chosen bijection. The source proposes it for $p$-solvable groups and investigates it computationally; no resolution is supplied.