  We explore the effects on nuclear bulk properties of using regularization
cutoffs larger than the nucleon mass within the chiral effective field theory
using a power counting that ensures order-by-order renormalization in the
two-nucleon system. To do so we calculate ground-state properties of the
$^{16}$O nucleus in the Hartree--Fock approach in a basis made up of plane
waves confined in a cube. We find a strong sensitivity to the regularization
cutoff through the counter-terms in attractive singular partial waves and to
the correction for spurious deeply bound states. This questions the possibility
of testing in nuclei the renormalization-group invariance of renormalizable
potentials from chiral effective field theory at leading order. A possible way
out of this problem is proposed.
