Can we prove the "Bertrand postulate" for primes $a \pmod q$, namely: there is always a prime number $p\equiv a \pmod q$ betwen $nq$ and $nq^2$ for every $n>0$ and $(a,q)=1$. (This would mean that between $nq$ and $nq^2$ there are always at least $\varphi(q)$ primes, each belonging to a different residue class modulo $q$.)