In recent years, several claims of detection of weak harmonic signals in the sub-tidal band of frequencies of high-quality gravimeter data have been made. Here, we review our attempts at confirming Smylie's claim of detection of the Slichter mode of inner-core oscillation using the same four data sets he and his colleagues used. We further examine the homogeneous 2 year data sets obtained from the superconducting gravimeters sited in Strasbourg and at Cantley, Quebec. We show that the power spectra of tidal-reduced, pressure and slew-corrected gravity residuals from these data are indistinguishable from a random walk process except in narrow bands dominated by residual earth tides and by harmonics of the diurnal thermal-atmospheric tide. Such a random 'brown' residual could result from mechanical instabilities and electronic noise in the instruments themselves, from site-specific tectonic noise, from local hydrological variations and pier instabilities or it could derive from unaccounted for variability in the atmospheric pressure in the vicinity of the instruments. We argue that the homogeneous 2 year data sets show no evidence whatever of Slichter harmonics even though our preprocessing methods reduce 'apparent noise levels' in the core-mode band by almost an order magnitude in comparison with the four data sets originally employed by Smylie et al. and, in their first attempted confirmation of his claimed discovery, by Jensen et al.