Feit's conjecture on character conductors and element orders

Let $G$ be a finite group and let $n$ be a positive integer. An irreducible character of $G$ has conductor $n$ when its character field has conductor $n$. Feit's conjecture. If $G$ has an irreducible character of conductor $n$, then $G$ has an element of order $n$. Here the conductor of a character is the conductor of the cyclotomic field generated by its character values. The conjecture would imply Brauer's Problem 41 and several earlier results; the supplied text does not establish whether it is resolved.