Evolution of the BH mass for the ‘EddLim’ (dotted blue), ‘sEdd’ (solid orange), ‘sEddThm’ (densely dash-dotted violet) and ‘sEddNF’ simulations (dash-dotted green). Both ‘sEdd’ and ‘sEddThm’ simulations start at 206.4 Myr (colored filled semi-circles). In dashed red is also added the expected evolution of a 10⁶ M_⊙ BH if it accreted constantly at the Eddington limit. It is clear that super-Eddington accretion, coupled with feedback does not help the BH to grow. We note that without jets (i.e. ‘sEddThm’ run), BHs with super-Eddington accretion are less affected by their feedback and are able to grow above the limit for at least 60 Myr. In fact, the BH reaches a higher mass than the one in ‘EddLim’ (see Sect. 4.2 for more details).