  We observe quasi-static incommensurate magnetic peaks in neutron scattering
experiments on layered cobalt oxides La2-xSrxCoO4 with high Co oxidation states
that have been reported to be paramagnetic. This enables us to measure the
magnetic excitations in this highly hole-doped incommensurate regime and
compare our results with those found in the low-doped incommensurate regime
that exhibit hourglass magnetic spectra. The hourglass shape of magnetic
excitations completely disappears given a high Sr doping. Moreover, broad
low-energy excitations are found, which are not centered at the incommensurate
magnetic peak positions but around the quarter-integer values that are
typically exhibited by excitations in the checkerboard charge ordered phase.
Our findings suggest that the strong inter-site exchange interactions in the
undoped islands are critical for the emergence of hourglass spectra in the
incommensurate magnetic phases of La2-xSrxCoO4.
