  In this work we revisit the growth of small primordial black holes (PBHs)
immersed in a quintessential field and/or radiation to the supermassive black
hole (SMBHs) scale. We show the difficulties of scenarios in which such huge
growth is possible. For that purpose we evaluated analytical solutions of the
differential equations (describing mass evolution) and point out the strong
fine tuning for that conclusions. The timescale for growth in a model with a
constant quintessence flux is calculated and we show that it is much bigger
than the Hubble time.The fractional gain of the mass is further evaluated in
other forms, including quintessence and/or radiation. We calculate the
cosmological density $\Omega$ due to quintessence necessary to grow BHs to the
supermassive range and show it to be much bigger than one. We also describe the
set of complete equations analyzing the evolution of the BH+quintessence
universe, showing some interesting effects such the quenching of the BH mass
growth due to the evolution of the background energy. Additional constraints
obtained by using the Holographic Bound are also described. The general
equilibrium conditions for evaporating/accreting black holes evolving in a
quintessence/radiation universe are discussed in the Appendix.
