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Super-entropic black holes possess finite-area but noncompact event horizonsand violate the reverse isoperimetric inequality. It has been conjectured thatsuch black holes always have negative specific heat at constant volume $C_{V}$or negative specific heat at constant pressure $C_{P}$ whenever $C_{V}>0$,making them unstable in extended thermodynamics. In this paper, we test thisinstability conjecture on a family of nonlinear electrodynamics black holes,namely $3$D Einstein-Born-Infeld (EBI) AdS black holes. Our results show thatwhen nonlinear electrodynamics effects are weak, the instability conjecture isvalid. However, the conjecture can be violated in some parameter region whennonlinear electrodynamics effects are strong enough. This observation thusprovides a counter example to the instability conjecture, which suggests thatsuper-entropic black holes could be thermodynamically stable.

Thermodynamic Instability of $\mathrm{{3D}}$ Einstein-Born-Infeld AdS Black Holes