  We show that the nonlocal two-flavor Nambu--Jona-Lasinio model predicts the
enhancement of both chiral and axial symmetry breaking as the chiral imbalance
of hot QCD matter, regulated by a chiral chemical potential $\mu_5$, increases.
The two crossovers are reasonably close to each other in the range of $\mu_5$
examined here and the pseudocritical temperatures rise with $\mu_5$. The
curvatures of the chiral and axial crossovers for the chiral quark chemical
potential approximately coincide and give $\kappa_5 \simeq - 0.011$. We point
out that the presence of $\mu_5$ in thermodynamic equilibrium is inconsistent
with the fact that the chiral charge is not a Noether-conserved quantity for
massive fermions. The chiral chemical potential should not, therefore, be
considered as a true chemical potential that sets a thermodynamically stable
environment in the massive theory, but rather than as a new coupling that may
require a renormalization in the ultraviolet domain. The divergence of an
unrenormalized chiral density, \corr{coming from zero-point fermionic
fluctuations,} is a consequence of this property. We propose a solution to this
problem via a renormalization procedure.
