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The outstanding mass growth of supermassive black holes (SMBHs) at theReionisation Epoch and how it is related to the concurrent growth of their hostgalaxies, poses challenges to theoretical models aimed at explaining how thesesystems formed in short timescales (<1 Gyr). To trace the average evolutionarypaths of quasi-stellar objects (QSOs) and their host galaxies in the BHmass-host mass ($M_{\rm dyn}$) plane, we compare the star formation rate (SFR),derived from the accurate estimate of the dust temperature and the dust mass($T_{\rm dust}, M_{\rm dust}$), with the BH accretion rate. To this aim, weanalysed a deep, $900$ pc resolution ALMA observation of the sub-mm continuum,[CII] and H$_2$O of the $z\sim 6$ QSO J2310+1855, enabling a detailed study ofdust properties and cold gas kinematics. We performed an accurate SED analysisobtaining a dust temperature of $T_{\rm dust} = 71$ K and a dust mass of$M_{\rm dust}= 4.4 \times 10^8\ \rm M_{\odot}$. The implied AGN-corrected SFRis $1240 \ \rm M_{\odot}yr^{-1}$, a factor of 2 smaller than previouslyreported for this QSO. We derived the best estimate of the dynamical mass$M_{\rm dyn} = 5.2\times 10^{10}\ \rm M_{\odot}$ within $r = 1.7$ kpc, based ona dynamical model of the system. We found that ${\rm SFR}/M_{\rm dyn}>\dotM_{\rm BH}/M_{\rm BH}$, suggesting that AGN feedback might be efficientlyacting to slow down the SMBH accretion, while the stellar mass assembly isstill vigorously taking place in the host galaxy. In addition, we were alsoable to detect high-velocity emission on the red and blue sides of the [CII]emission line, that traces a gaseous outflow, and for the first time, we mappeda spatially-resolved water vapour disk through the H$_2$O v=0$3_{(2,2)}-3_{(1,3)}$ emission line detected at $\nu_{\rm obs} = 274.074$ GHz,whose kinematic properties and size are broadly consistent with those of the[CII] disk.

The black hole and host galaxy growth in an isolated $z\sim 6$ QSO observed with ALMA