  The best-motivated scenario for a sizable primordial black hole (PBH)
contribution to the LIGO/Virgo binary black hole mergers invokes the QCD phase
transition, which naturally enhances the probability to form PBH with masses of
stellar scale. We reconsider the expected mass function associated not only to
the QCD phase transition proper, but also the following particle antiparticle
annihilation processes, and analyse the constraints on this scenario from a
number of observations: The specific pattern in cosmic microwave background
(CMB) anisotropies induced by accretion onto PBHs, CMB spectral distortions,
gravitational wave searches, and direct counts of supermassive black holes
(SMBHs) at high redshift. We find that the scenario is not viable, unless an ad
hoc mass evolution for the PBH mass function and a cutoff in power-spectrum
very close to the QCD scale are introduced by hand. Despite these negative
results, we note that a future detection of coalescing binaries involving
sub-solar PBHs has the potential to check the cosmological origin of SMBHs at
the $e^\pm$ annihilation epoch, if indeed the PBH mass function is shaped by
the changes to the equation of state driven by the thermal history of the
universe.
