  We argue that in theories of quantum gravity with discrete gauge symmetries,
e.g. $\textbf{Z}_k$, the gauge couplings of U$(1)$ gauge symmetries become weak
in the limit of large $k$, as $g\to k^{-\alpha}$ with $\alpha$ a positive order
1 coefficient. The conjecture is based on black hole arguments combined with
the Weak Gravity Conjecture (or the BPS bound in the supersymmetric setup), and
the species bound. We provide explicit examples based on type IIB on
AdS$_5\times \textbf{S}^5/\textbf{Z}_k$ orbifolds, and M-theory on
AdS$_4\times\textbf{S}^7/\textbf{Z}_k$ ABJM orbifolds (and their type IIA
reductions). We study AdS$_4$ vacua of type IIA on CY orientifold
compactifications, and show that the parametric scale separation in certain
infinite families is controlled by a discrete $\textbf{Z}_k$ symmetry for
domain walls. We accordingly propose a refined version of the strong AdS
Distance Conjecture, including a parametric dependence on the order of the
discrete symmetry for 3-forms.
