  The $M_{\bullet} - \sigma$ relation establishes a connection between central
black holes (BHs) and their host spheroids. Supported by observations at
$M_{\bullet} \gtrsim 10^5 \, M_{\odot}$, there is limited data on its validity
at lower masses. Employing a semi-analytical model to simulate the combined
evolution of BHs and their host galaxies, we predict the observational
consequences of assuming a bimodality in the accretion efficiency of BHs, with
low-mass BHs ($M_{\bullet} \lesssim 10^5 \, M_{\odot}$) accreting
inefficiently. We predict a departure from the $M_{\bullet} - \sigma$ relation
at a transitional BH mass $\sim 10^5 \, M_{\odot}$, with lower-mass BHs unable
to reach the mass dictated by the relation and becoming disconnected from the
evolution of the host galaxy. This prediction is an alternative to previous
works suggesting a flattening of the relation at $\sim 10^5-10^6 \, M_{\odot}$.
Furthermore, we predict a deficit of BHs shining at bolometric luminosities
$\sim 10^{42} \, \mathrm{erg \, s^{-1}}$. Joined with a detection bias, this
could partly explain the scarce number of intermediate-mass BHs detected.
Conversely, we predict an increase in source density at lower bolometric
luminosities, $<10^{42} \, \mathrm{erg \, s^{-1}}$. Because our predictions
assume a bimodal population of high-redshift BH seeds, future observations of
fainter BHs will be fundamental for constraining the nature of these seeds.
