Precision mass measurements in the zirconium region pin down the mass surface across the neutron midshell at $N=66$

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.