No common rational multiple for all Dirichlet ordinates

Let a Dirichlet ordinate be a nonnegative real number that is a $\chi$-ordinate for some Dirichlet character $\chi$. Silberman's consequence of linear independence. There does not exist a real number $\omega$ such that every Dirichlet ordinate is a rational multiple of $\omega$. The source attributes this observation to Silberman and presents it as a seemingly simple consequence of the linear-independence conjecture; it is therefore open with that conjectural status.