Do Dirichlet $L$-functions with greater modulus have a greater number of zeros lying off the critical line?

Question

Ignoring the GRH, for $\sigma>\frac{1}{2}$, is it the case that $$\sum_{primitive\chi\bmod{q}}N(\sigma,T,\chi)\geq\sum_{primitive\chi\bmod{d}}N(\sigma,T,\chi)$$ if $d|q$. Of course, the zero density estimates available do not contradict this statement, but I was wondering if the statement is known to be true.