The optimal upper-bound conjecture for Farey sequences of $\Gamma_0(p^n)$

Let $p$ be a prime and let $n\geq 3$. A Farey sequence of $\Gamma_0(p^n)$ is denoted by $\{e_i\}$. Optimal upper-bound conjecture. The group $\Gamma_0(p^n)$ has a Farey sequence $\{e_i\}$ such that

$$e_i\leq p^{n-1}.$$

The preceding algorithm achieves the analogous optimal upper bound $2^{n-1}$ for $\Gamma_0(2^n)$, motivating the expectation that a suitable modification works for $\Gamma_0(p^n)$ in many cases; the asserted bound for all primes $p$ and $n\geq 3$ is not established here.