Linear growth of strong prime-to-p weight bounds

Determine whether, for every prime p, there exists a constant C_p greater than zero such that the least strong prime-to-p weight bound B^(p)(1,p,m) for level-one eigenforms satisfies B^(p)(1,p,m) ≤ C_p φ(p^m) for every integer m ≥ 1.

Background

The paper derives explicit strong prime-to-p weight bounds for level-one eigenforms for several prime powers and compares them with the Euler totient φ(p^m). The observed values are of the same general order as φ(p^m), but the data do not yield a uniform formula valid for all primes and exponents.

The question asks whether this observed growth can be bounded uniformly in the exponent m by a constant depending only on the prime p. It concerns the asymptotic behavior of the least bound for congruences away from p, rather than the stronger full Hecke-eigensystem problem.