Aims. The interplay between the accretion of supermassive black holes (SMBHs) and the stellar mass growth of the host galaxies is still a matter of hot debate. The accretion of the central SMBHs is expected to release energy under the form of active galactic nuclei. This energy is believed to impact the star formation activity and contribute to the quenching of the host galaxies. Here, we address this key unsolved issue with our cosmological semi-empirical model DECODE (Discrete statistical sEmi-empiriCal mODEl). Methods. In DECODE, we grow galaxies with their star formation rate linked to halo accretion rate distributions via abundance matching. SMBHs are evolved following the stellar mass growth of their host galaxies by assigning an accretion rate at each redshift from the empirical Eddington ratio distributions and duty cycles. We tested the assumption that galaxies permanently quench when their central SMBHs approach the limit imposed by the observed MBH − σ★ relation, as a proxy of SMBH disruptive feedback. Results. We find that simply imposing the MBH − σ★ condition is sufficient to generate a fraction of quenched galaxies consistent with current data, including the newest ones from Euclid. In addition, our minimal data-driven model also predicts SMBH scaling relations consistent in slope and normalisation with those that have been observed, and an MBH − M★ relation weakly evolving with redshift. The model also naturally generates SMBH accretion rates peaking within 1 Gyr of their host star formation histories. Interestingly, we note that all the main predictions on galaxy quenched fractions and SMBH growth histories and scaling relations are degenerate with those expected in a halo quenching model. Conclusions. The comprehensive data-driven model presented in this work represents an invaluable tool to investigate SMBH demography across time and environments in an accurate, physically motivated manner, ideally suited to rapidly exploring the implications from large surveys, such as Euclid and Rubin-LSST.
