  High-precision mass measurements of exotic $^{95-97}$Ag isotopes close to the
$N = Z$ line have been conducted with the JYFLTRAP double Penning trap mass
spectrometer, with the silver ions produced using the recently commissioned
inductively-heated hot cavity catcher laser ion source at the Ion Guide Isotope
Separator On-Line facility. The atomic mass of $^{95}$Ag was directly
determined for the first time. In addition, the atomic masses of
$\beta$-decaying 2$^+$ and 8$^+$ states in $^{96}$Ag have been identified and
measured for the first time, and the precision of the $^{97}$Ag mass has been
improved. The newly measured masses, with a precision of $\approx$ 1 keV/c$^2$,
have been used to investigate the $N =$ 50 neutron shell closure confirming it
to be robust. Empirical shell-gap and pairing energies determined with the new
ground-state mass data are compared with the state-of-the-art \textit{ab
initio} calculations with various chiral effective field theory Hamiltonians.
The precise determination of the excitation energy of the $^{96m}$Ag isomer in
particular serves as a benchmark for \textit{ab initio} predictions of nuclear
properties beyond the ground state, specifically for odd-odd nuclei situated in
proximity to the proton dripline below $^{100}$Sn. In addition, density
functional theory (DFT) calculations and configuration-interaction shell-model
(CISM) calculations are compared with the experimental results. All theoretical
approaches face challenges to reproduce the trend of nuclear ground-state
properties in the silver isotopic chain across the $N =$50 neutron shell and
toward the proton drip-line.
