Hung–Tiep's conductor-degree conjecture for character fields

Let $G$ be a finite group and let $\chi$ be an irreducible complex character of $G$. Let $\mathbb{Q}(\chi)$ be the field generated by the values of $\chi$, and let $c(\chi)$ be the smallest positive integer such that $\mathbb{Q}(\chi)\subseteq\mathbb{Q}_{c(\chi)}$, where $\mathbb{Q}_n$ is the field generated by a primitive $n$th root of unity. Hung–Tiep's conjecture. One has

$$|\mathbb{Q}_{c(\chi)}: \mathbb{Q}(\chi)| \leq \chi(1).$$

Hung and Tiep verified this for alternating, unitary, and general linear groups, and whenever $\chi(1)\leq 3$; Hung, Tiep, and Zalesski verified it when $\chi(1)$ is prime. The general case is presented here as a conjecture, although the paper's abstract states that the corresponding question is answered negatively at the element level.