Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number of prime numbers $p \leq n$ in the residue class $r$ (mod $m$). Further let $1 = r_1 < r_2 < \dots < r_{\varphi(m)} = m-1$ be the prime residues (mod $m$), and let $\sigma \in {\rm Sym}(\{r_1, \dots, r_{\varphi(m)}\})$ be a permutation of them. Are there always infinitely many $n \in \mathbb{N}$ such that $\pi(m,\sigma(r_1),n) < \pi(m,\sigma(r_2),n) < \dots < \pi(m,\sigma(r_{\varphi(m)}),n)$?

If the answer is *yes*, are explicit bounds $B(m)$ known such that
for every permutation $\sigma$ of the prime residues (mod $m$) there is
*at least one* such $n$ less than $B(m)$?

If the answer is *no*, for which permutations $\sigma$ are there infinitely many
such $n$, and for which are not?