Overmassive black holes in the $M_{\rm BH} - \sigma$ diagram do not belong to over (dry) merged galaxies

Semi-analytical models in a $\Lambda$CDM cosmology have predicted the presence of outlying, "overmassive" black holes at the high-mass end of the (black hole mass -- galaxy velocity dispersion) $M_{\rm BH} - \sigma$ diagram (which we update here with a sample of 89 galaxies). They are a consequence of having experienced more dry mergers -- thought not to increase a galaxy's velocity dispersion -- than the "main-sequence" population. Wet mergers and gas-rich processes, on the other hand, preserve the main correlation. Due to the scouring action of binary supermassive black holes, the extent of these dry mergers (since the last significant wet merger) can be traced by the ratio between the central stellar mass deficit and the black hole mass ($M_{\rm def,*} / M_{\rm BH}$). However, in a sample of 23 galaxies with partially depleted cores, including central cluster galaxies, we show that the "overmassive" black holes are actually hosted by galaxies that appear to have undergone the lowest degree of such merging. In addition, the rotational kinematics of 37 galaxies in the $M_{\rm BH} - \sigma$ diagram reveals that fast and slow rotators are not significantly offset from each other, also contrary to what is expected if these two populations were the product of wet and dry mergers respectively. The observations are thus not in accordance with model predictions and further investigation is required.