Now Gadotti & Kauffmann have presented an analysis of SDSS data that appears to support this conclusion. This plot shows the bulge mass vs. velocity dispersion for ellipticals, classical bulges, and pseudo-bulges. The ellipticals follow a tight relation, and the classical bulges also follow a fairly tight relation but with an offset. However the pseudo-bulges don't seem to follow much of a relation, tend to have significantly lower masses than ellipticals/classical bulges at a fixed velocity dispersion.

If pseduo-bulges don't follow the same M_bulge-sigma relation as other galaxies, then they can't follow both the standard M_bh-M_bulge relation and the M_bh-sigma relation at the same time. Perhaps pseudo-bulges follow only one of these relations (as suggested by the Greene et al. paper), or maybe they follow neither.

One possible explanation mentioned by the authors for the observation that pseudo-bulges don't follow the same M_bulge-sigma relation is that bars (which may be so small as to be undetected) artificially enhance the observed sigma. Another is that pseudo-bulges aren't relaxed, so the virial theorem doesn't apply.