What sets the black hole mass versus bulge velocity dispersion relation?
In plain words
The mass of a galaxy's central black hole tracks the random speeds of stars in its central bulge with surprisingly little scatter, although the black hole is far too small to affect those stars by gravity. Either the black hole's energy output regulates the galaxy, or repeated mergers average the two masses together.
Precise statement
Classical bulges and ellipticals follow $M_{\mathrm{BH}} \sim \sigma^{\beta}$ with $\beta \sim 4-5$ and intrinsic scatter ~ 0.3 dex in $\operatorname{log} M_{\mathrm{BH}}$, while pseudobulges and disks do not follow it tightly (Kormendy and Ho 2013, arXiv:1304.7762). Determine the origin: self-regulation by AGN momentum- or energy-driven winds (predicting a definite $\beta$ and normalization), central-limit averaging through successive mergers, or a common gas supply. The answer is a mechanism whose predicted redshift evolution of slope, normalization and scatter matches measurements.
What would settle it
Measurement of the $M_{\mathrm{BH}}-\sigma$ slope, normalization and scatter versus redshift with black hole masses independent of virial line-width estimates, compared with the distinct predictions of feedback and merger-averaging models.