  The construction of black hole spacetimes that are regular (singularity-free)
is plagued by the "mass inflation" instability, a classical perturbation
instability induced by the surface gravity at the inner horizon and
characterized by exponentially diverging stress-energy there. Recently, a class
of "inner-extremal" regular black holes was proposed that possesses a vanishing
inner-horizon surface gravity and therefore avoids mass inflation, while still
maintaining a horizon separation and a non-zero outer-horizon surface gravity.
However, when semiclassical effects are taken into account, it is found that an
inner-horizon instability remains for generic inner-extremal regular black
holes formed from collapse. This semiclassical divergence is analyzed from the
perspective of both the effective Hawking temperature and the renormalized
stress-energy tensor, and its origin and genericity are examined in detail.
