Parker's conjecture on odd orthogonal determinants

Let $G$ be a finite group and let $\chi$ be an orthogonally stable character. Write $\mathbb{Q}(\chi)$ for its field of character values, and call the square class of a positive rational number odd when its unique squarefree positive integral representative is odd.

Parker's conjecture. If $\mathbb{Q}(\chi)=\mathbb{Q}$, then $\det(\chi)$ is odd.

This conjecture arose from extensive computations of orthogonal determinants of finite groups. The supplied text gives no resolution, so its status is recorded as open.