  We have successfully grown centimeter-sized layered $R$SrNiO$_4$ single
crystals under high oxygen pressures of 120 bar by the floating zone technique.
This enabled us to perform neutron scattering experiments where we observe
close to quarter-integer magnetic peaks below $\sim$77 K that are accompanied
by steep upwards dispersing spin excitations. Within the high-frequency Ni-O
bond stretching phonon dispersion, a softening at the propagation vector for a
checkerboard modulation can be observed. Together with our spin wave
simulations these observations reveal that this Ni$^{3+}$ system exhibits
charge disproportionation with charges segregating into a checkerboard pattern
within a nano phase separation scenario rather than showing a Jahn-Teller
effect.
