  Super-entropic black holes possess finite-area but noncompact event horizons
and violate the reverse isoperimetric inequality. It has been conjectured that
such 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 this
instability conjecture on a family of nonlinear electrodynamics black holes,
namely $3$D Einstein-Born-Infeld (EBI) AdS black holes. Our results show that
when nonlinear electrodynamics effects are weak, the instability conjecture is
valid. However, the conjecture can be violated in some parameter region when
nonlinear electrodynamics effects are strong enough. This observation thus
provides a counter example to the instability conjecture, which suggests that
super-entropic black holes could be thermodynamically stable.
