Galois-invariant height-zero characterization of normal Sylow subgroups

Let $p$ be a prime, and let $\mathcal J_p$ be the subgroup of $\operatorname{Gal}({\mathbb Q}^{\rm ab}/{\mathbb Q})$ consisting of automorphisms of order $p$ that fix all $p$-power-order roots of unity. For a finite group $G$, let ${{\operatorname{Irr}}}_{\mathcal J_p}(B_p(G))$ be the set of $\mathcal J_p$-invariant irreducible characters of the principal $p$-block $B_p(G)$.

Galois-invariant normal Sylow subgroup conjecture. Let $G$ be a finite group and let $q$ be a prime. Then $G$ has a normal Sylow $q$-subgroup if and only if, for any $p\neq q$ that divides $|G|$, $q$ does not divide $\chi(1)$ for every $\chi\in{{\operatorname{Irr}}}_{\mathcal J_p}(B_p(G))$.

This is proposed as a Galois-equivariant strengthening of the height-zero character criterion. The paper presents it as a further direction, and no resolution is given.