  We investigate the asymptotic Tower Weak Gravity Conjecture in weak coupling
limits of open string theories with minimal supersymmetry in four dimensions,
focusing for definiteness on gauge theories realized on 7-branes in F-theory.
Contrary to expectations, we find that not all weak coupling limits contain an
obvious candidate for a tower of states marginally satisfying the
super-extremality bound. The weak coupling limits are classified geometrically
in the framework of EFT string limits and their generalizations. We find three
different classes of weak coupling limits, whose physics is characterized by
the ratio of the magnetic weak gravity scale and the species scale. The
four-dimensional Tower Weak Gravity Conjecture is satisfied by the (non-BPS)
excitations of the weakly coupled EFT string only in emergent string limits,
where the EFT string can be identified with a critical (heterotic) string. All
other weak coupling limits lead to a decompactification either to an in general
strongly coupled gauge theory coupled to gravity or to a defect gauge theory
decoupling from the gravitational bulk, in agreement with the absence of an
obvious candidate for a marginally super-extremal tower of states.
