  The Weak Gravity Conjecture has recently been re-formulated in terms of a
particle with non-negative self-binding energy. Because of the dual conformal
field theory (CFT) formulation in the anti-de Sitter space the conformal
dimension $\Delta (Q)$ of the lowest-dimension operator with charge Q under
some global U(1) symmetry must be a convex function of Q. This property has
been conjectured to hold for any (unitary) conformal field theory and
generalized to larger global symmetry groups. Here we refine and further test
the convex charge conjecture via semiclassical computations for fixed charge
sectors of different theories in different dimensions. We analyze the convexity
properties of the leading and next-to-leading order terms stemming from the
semiclassical computation, de facto, extending previous tests beyond the
leading perturbative contributions and to arbitrary charges. In particular, the
leading contribution is sufficient to test convexity in the semiclassical
computations. We also consider intriguing cases in which the models feature a
transition from real to complex conformal dimensions either as a function of
the charge or number of matter fields. As a relevant example of the first kind,
we investigate the $O(N)$ model in $4+\epsilon$ dimensions. As an example of
the second type we consider the $U(N)\times U(M)$ model in $4-\epsilon$
dimensions. Both models display a rich dynamics where, by changing the number
of matter fields and/or charge, one can achieve dramatically different physical
regimes. We discover that whenever a complex conformal dimension appears, the
real part satisfies the convexity property.
