$p$-rational characters and self-normalizing Sylow $p$-subgroups

Let $G$ be a finite group, $p$ a prime, and $P$ a Sylow $p$-subgroup of $G$. Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of $p'$-degree of $G$ and the irreducible characters of $p'$-degree of $\mathbf{N}_G(P)$, which preserves field of values of correspondent characters (over the $p$-adics). This strengthening of the McKay conjecture has several consequences. In this paper we prove one of these consequences: If $p>2$, then $G$ has no non-trivial $p'$-degree $p$-rational irreducible characters if and only if $\mathbf{N}_G(P)=P$.