  The occurrence of pair-instability supernovae is predicted to prevent the
formation of black holes with masses $\gtrsim 50 M_\odot$. Recent
gravitational-wave detections in this mass range require an explanation beyond
that of standard stellar collapse. Current modeling strategies include the
hierarchical assembly of previous generations of black-hole mergers as well as
other mechanisms of astrophysical nature (lowered nuclear-reaction rates,
envelope retention, stellar mergers, accretion, dredge-up episodes). In this
paper, we point out the occurrence of an exclusion region that cannot be easily
populated by hierarchical black-hole mergers. A future gravitational-wave
detection of a black hole with mass $\gtrsim 50M_\odot$ and spin $\lesssim 0.2$
will indicate that the pair-instability mass gap is polluted in some other way.
Such a putative outlier can be explained using hierarchical mergers only with
considerable fine-tuning of both mass ratio and spins of the preceding
black-hole merger -- an assumption that can then be cross-checked against the
bulk of the gravitational-wave catalog.
