Observations of high-redshift quasars provide information on the massive black holes (MBHs) powering them and the galaxies hosting them. Current observations of $z gtrsim 6$ hosts, at sub-mm wavelengths, trace the properties of cold gas, and these are used to compare with the correlations between MBHs and galaxies characterising the $z=0$ population. The relations at $z=0$, however, rely on stellar-based tracers of the galaxy properties. We perform a very-high resolution cosmological zoom-in simulation of a $z=7$ quasar including state-of-the-art non-equilibrium chemistry, MBH formation, growth and feedback, to assess the evolution of the galaxy host and the central MBH, and compare the results with recent ALMA observations of high-redshift quasars. We measure both the stellar-based quantities used to establish the $z=0$ correlations, as well as the gas-based quantities available in $z gtrsim 6$ observations, adopting the same assumptions and techniques used in observational studies. The high-redshift studies argued that MBHs at high redshift deviate from the local MBH-galaxy correlations. In our analysis of the single galaxy we evolve, we find that the high-redshift population sits on the same correlations as the local one, when using the same tracers used at $z=0$. When using the gas-based tracers, however, MBHs appear to be over-massive. The discrepancy between local and high-redshift MBHs seems caused by the different tracers employed, and necessary assumptions, and not by an intrinsic difference. Better calibration of the tracers, higher resolution data and availability of facilities that can probe the stellar population will be crucial to assess precisely and accurately high-redshift quasar hosts.
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