In 2006, Arkani-Hamed, Motl, Nicolis, and Vafa presented evidence that gravity is the weakest force, a claim that would often be called the "weak gravity conjecture" (WCG) later. Yes, the swampland on Figure 1, so different from the peaceful Czech landscape, is some territory between Poland, Ukraine, and Belarus. ;-) It means that for any non-gravitational force, imagine the electrostatic or magnetostatic forces, there have to exist light enough particle species that are sufficiently charged so that their mutual non-gravitational forces exceed the gravitational ones.
Be sure that this hypothesis passes the experimental test – gravity is the weakest force in the world around us. But the point is that it couldn't have been otherwise.
This claim – an inequality of a sort – seems to be satisfied everywhere in string theory. Some partial proofs for classes of stringy vacua may be at least sketched. Moreover, even when you ignore any string constructions, the inequality is needed to avoid the black hole remnants which would make any theory of quantum gravity inconsistent. For remnants not to occur, extremal black holes have to be able to decay as well, and by charge and mass conservation laws, that's only possible if the mass/charge ratio of some of the Hawking particles they emit obeys the opposite inequality than the allowed black holes – when the Hawking particles act as "supraextremal black holes" when it comes to their charge/mass ratio.
I am sure that a rule like that is true but I am equally sure that we haven't found the most accurate formulation of the inequality in the most general vacuum of quantum gravity. A related problem is that we couldn't find the most "primordial" reasons behind the claim. We couldn't even figure out whether the inequality is truly fundamental and far-reaching – like the Heisenberg uncertainty principle (although this is almost certainly too formidable a foe to beat) or just some minor technical result.
A derivation of the inequality from some reliable mathematics that is more tightly connected with the formalism of quantum field theory would always clarify the situation. The first hep-th paper today is
The result that the authors are most proud about is the inequality (4)\[
a_1+b_1 -b_3 +c_1 +c_2 +3c_3 \geq 0
\] but they have two similar inequalities and some extra comments. I must tell you that the constants \(a_i,b_i,c_i\) are coefficients of four-derivative terms in the photon-graviton (extended Maxwell-Einstein) action. The letters \(a\) refer to \(F^4\) terms, \(b\) are the mixed \(F^2 R\) terms, and the \(c\) constants are the purely gravitational \(R^2\) terms. The indices of \(a,b,c\) encode different ways how the Lorentz indices of \(F,R\) may be contracted.
I am a bit confused why some of these coefficients aren't obliged to be positive definite (or negative definite) separately from others, but I am probably being stupid. If they know what they're doing, the known arguments in quantum field theory allowed pretty much any values of all these \(a,b,c\) parameters but they have derived new constraints that look incomprehensible as mathematics but they can interpret them as a manifestation of the weak gravity conjecture (it's not trivial to understand why there is a relationship at all but I hope it can be understood at the end).
The weak gravity conjecture became one of the most well-known examples of Cumrun Vafa's "swampland paradigm" – string theory or a consistent theory of quantum gravity (these are really two phrases denoting the same thing – just the first one makes you think more constructively and the second one more generally, bootstrappily) makes lots of universal predictions and constraints that hold even if one is ignorant about the vacuum that Nature chose, constraints that used to be unknown to generic builders of effective quantum field theories.
Unitarity and analyticity that were used as a starting point are fundamental in principle but their consequences for a theory always look like human-unfriendly technical mess. Maybe we need to understand all these features at a more intuitive level to get more familiar with the true foundations of quantum gravity.