Conundrum of regular black holes with nonlinear electromagnetic fields

The search for regular black holes with nonlinear electromagnetic fields has sprouted numerous candidates, each exhibiting certain virtues but often accompanied by significant drawbacks. We demonstrate that Komar mass, electric charge and magnetic charge are mutually dependent in regular black holes with nonlinear electromagnetic fields, defined by a Lagrangian which is a function of both electromagnetic invariants, $F_{ab} F^{ab}$ and $F_{ab}{\star F}^{ab}$, regardless of the specific weak field limit of the theory. Also, we generalize one of the key no-go theorems by showing that static, spherically symmetric, electrically charged black holes in a theory respecting the relaxed Maxwellian weak field limit do not admit a bounded Kretschmann scalar. Finally, we address one of the long-standing niche questions, whether regular black hole solutions can exist when both electric and magnetic charges are present, by constructing an exotic family of regular dyonic black holes with nonlinear electromagnetic fields in theories respecting the Maxwellian weak field limit. Mounting evidence suggests that regularizing black holes through simplistic nonlinear extensions of Maxwell's electromagnetism entails a high cost in the form of unorthodox theoretical assumptions.