  Precision mass measurements of $^{104}$Y, $^{106}$Zr, $^{104,104m,109}$Nb,
and $^{111,112}$Mo have been performed with the JYFLTRAP double Penning trap
mass spectrometer at the Ion Guide Isotope Separator On-Line facility. The
order of the long-lived states in $^{104}$Nb was unambiguously established. The
trend in two-neutron separation energies around the $N=66$ neutron midshell
appeared to be steeper with respect to the Atomic Mass Evaluation 2020
extrapolations for the $_{39}$Y and $_{40}$Zr isotopic chains and less steep
for the $_{41}$Nb chain, indicating a possible gap opening around $Z=40$. The
experimental results were compared to the BSkG2 model calculations performed
with and without vibrational and rotational corrections. All of them predict
two low-lying minima for $^{106}$Zr. While the unaltered BSkG2 model fails to
predict the trend in two-neutron separation energies, selecting the more
deformed minima in calculations and removing the vibrational correction, the
calculations are more in line with experimental data. The same is also true for
the $2^+_1$ excitation energies and differences in charge radii in the Zr
isotopes. The results stress the importance of improved treatment of collective
corrections in large-scale models and further development of beyond-mean-field
techniques.
