  We analyze the largely accepted formulas for the asymptotic quasi-normal
frequencies of the non-extremal Reissner-Nordstr\"om black hole, (for the
electromagnetic-gravitational/scalar perturbations). We focus on the question
of whether the gap in the spacing in the imaginary part of the QNM frequencies
has a well defined limit as n goes to infinity and if so, what is the value of
the limit. The existence and the value of this limit has a crucial importance
from the point of view of the currently popular Maggiore's conjecture, which
represents a way of connecting the asymptotic behavior of the quasi-normal
frequencies to the black hole thermodynamics. With the help of previous results
and insights we will prove that the gap in the imaginary part of the
frequencies does not converge to any limit, unless one puts specific
constraints on the ratio of the two surface gravities related to the two
spacetime horizons. Specifically the constraints are that the ratio of the
surface gravities must be rational and such that it is given by two relatively
prime integers N, M, whose product is an even number. If the constraints are
fulfilled the limit of the sequence is still not guaranteed to exist, but if it
exists its value is given as the lowest common multiplier of the two surface
gravities. At the end of the paper we discuss the possible implications of our
results.
