  A class of Polyakov-loop-modified Nambu--Jona-Lasinio (PNJL) models have been
used to support a conjecture that numerical simulations of lattice-regularized
quantum chromodynamics (QCD) defined with a chiral chemical potential can
provide information about the existence and location of a critical endpoint in
the QCD phase diagram drawn in the plane spanned by baryon chemical potential
and temperature. That conjecture is challenged by conflicts between the model
results and analyses of the same problem using simulations of
lattice-regularized QCD (lQCD) and well-constrained Dyson-Schwinger equation
(DSE) studies. We find the conflict is resolved in favor of the lQCD and DSE
predictions when both a physically-motivated regularization is employed to
suppress the contribution of high-momentum quark modes in the definition of the
effective potential connected with the PNJL models and the four-fermion
coupling in those models does not react strongly to changes in the mean-field
that is assumed to mock-up Polyakov loop dynamics. With the lQCD and DSE
predictions thus confirmed, it seems unlikely that simulations of lQCD with
$\mu_5>0$ can shed any light on a critical endpoint in the regular QCD phase
diagram.
