on the evolution study.

We apply the fitting procedure described in Section 3.4 and use the fitting function expressed by Eq. \[11\]. The pivot points are equal to the values of Table 1 and the power of evolution, $\gamma$, is here fixed to the self-similar expectations introduced in Section 3. The results, in terms of best-fitting parameters and 1 $\sigma$ uncertainties, are reported in Tables 2 and 3 for all relations, ICM physics treatments, and for different redshifts.

The Slope
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The evolution of the slopes of the scaling relations, $\beta$, is shown in Fig. \[fig:3\] for all six scaling relations and ICM physics.

[**$M-{M_{\rm g}}$ relation.**]{} The slope of the $M-{M_{\rm g}}$ relation is approximately constant over time for the [[[nr]{}]{}]{}  and [[[csf]{}]{}]{} runs confirming the expectations: the total mass and the gas mass grow simultaneously. Indeed, dark matter and gas increase in mass by the same fraction when the mass growth happens via slow accretion (due to constant ratio between the densities of the gas and DM components in the cluster outskirts, e.g., @rasia.etal.2004) or via major merger (due to small dependence of the gas fraction on the total mass, e.g., @2013MNRAS.431.1487P [@eckert.etal.2015], and reference therein). For the 