  Recently, first-order phase transitions have been predicted as an effect of
the inclusion of quark anomalous magnetic moment (AMM) in the hot and
magnetized Nambu--Jona-Lasinio model (NJL). These transitions appear in the
chiral condensate for different combinations of AMM and magnetic fields and
could lead to inverse magnetic catalysis. However, in this work, we show that
the predicted first-order phase transitions are related to
regularization-dependent issues. To show this, we explore, in the context of
the vacuum magnetic regularization (VMR) scheme, two different scenarios: when
mass-dependent (MD) and mass-independent (MI) terms are present in the
subtraction of the divergences. In the MD case, as we increase the AMM value,
it is observed the appearance of a nonmassive minimum in the thermodynamical
potential, which induces a first-order phase transition from the massive
minimum. We argue that the MD terms must be avoided in order to satisfy the
predictions of Lattice QCD, and we propose a MI solution that is valid in the
limit which the magnetic fields are smaller than the squared of vacuum
effective quark mass.
