Left panel: black hole mass as a function of redshift (see Eq. (2)) for J1342 (red lines) and J1120 (green lines; these curves are related also to the source J0038 since the BH mass is similar to that of J1120). The evolutionary tracks are compared with the ones related to the sources SDSSJ013127.34–032100.1 (blue lines) and S5 0014+813 (orange lines). The final masses in the brackets correspond to the non-spinning BH computed for θ_v = 30° (for the case with a = 0.9982, the curves are similar and shifted rigidly to larger masses by a factor of ∼3.5). Solid lines refer to λ = 1, dashed to λ = 2, dotted to λ = 3; the radiative efficiency η = 0.1 is fixed: for J1342, J1120 and J0038, in the case with λ = 1 (ṁ = 10) a massive seed of ∼10⁴ M_⊙ is required at redshift z >  15 in order to reach the observed mass. In the case of λ = 2, the evolution could have begun with a seed of ∼10² M_⊙ at redshift z ∼ 13 − 14, growing exponentially in ∼0.4 Gyr. For λ = 3, a faster evolution of ∼0.2 Gyr could have begun at z ∼ 10 − 11. Instead for J013127 and 0014, if λ ∼ 1, a seed of ∼10² M_⊙ could have grown in ∼0.9 Gyr, starting at redshift z ∼ 15 and z ∼ 6, respectively. In the case with a fixed Eddington ratio λ = η ṁ ∼ 1, for the same value of ṁ, the radiative efficiency is smaller and the evolution is faster but the curves (not overplotted for clarity) show a similar trend. Right panel: zoom on the observed redshifts of J1342 (red lines) and J1120 (green lines) with the evolutionary tracks leading to black hole masses corresponding to the cases with a = 0 and a = 0.9982 (for J0038, the evolutionary tracks are similar to those of J1120, shifted to lower redshifts by 0.06, not overplotted for clarity). Solid lines describe the case with λ = 1 and dashed lines the case with λ = 2. At redshift z = 7.54, J1120 (J0038) could have been more (or less) massive than J1342 depending on the values of λ and the BH spin: for example, in the case with λ = 2, J1120 (J0038) was less massive than J1342 by a factor ≳2 for any spin value; with λ = 1, J1120 (J0038) was more massive than J1342 only if the sources had different spins.