  We prove a version of the Weak Gravity Conjecture for 6d F-theory or
heterotic string compactifications with 8 supercharges. This sharpens our
previous analysis by including massless scalar fields. The latter are known to
modify the Weak Gravity Conjecture bound in two a priori independent ways:
First, the extremality condition of a charged black hole is modified, and
second, the test particles required to satisfy the Weak Gravity Conjecture are
subject to additional Yukawa type interactions. We argue on general grounds
that at weak coupling, the two types of effects are equivalent for a tower of
asymptotically massless charged test particles predicted by the Swampland
Distance Conjecture. We then specialise to F-theory compactified on elliptic
Calabi-Yau three-folds and prove that the precise numerical bound on the
charge-to-mass ratio is satisfied at weak coupling. This amounts to an
intriguing coincidence of two a priori different notions of extremality, namely
one based on the balance of gauge, gravitational and scalar forces for extremal
(non-BPS) black holes, and the other encoded in the modular properties of
certain Jacobi forms. In the presence of multiple abelian gauge group factors,
the elliptic genus counting these states is a lattice quasi-Jacobi form of
higher rank, and we exemplify this in a model with two abelian gauge group
factors.
