  Motivated by the weak gravity conjecture, arXiv:2108.04594 conjectured that
in any CFT, the minimal operator dimension at fixed charge is a convex function
of the charge. In this letter we construct a counterexample to this convexity
conjecture, which is a clockwork-like model with some modifications to make it
a weakly-coupled CFT. We also discuss further possible applications of this
model and some modified versions of the conjecture which are not ruled out by
the counterexample.
