Parker's odd orthogonal discriminant conjecture

Let $G$ be a finite group and let $\chi$ be an orthogonally stable rational character of $G$. Its orthogonal discriminant is the square class represented by a unique square-free integer $d$. Parker's conjecture. The integer $d$ is odd.

Orthogonal discriminants encode the square class of the discriminant of non-degenerate $G$-invariant quadratic forms affording an orthogonally stable character. The conjecture is motivated by computations of thousands of orthogonal discriminants of orthogonally stable characters of finite simple groups, in which no even representative was found; its general status is not specified in the source.