  The theory for single stellar evolution predicts a gap in the mass
distribution of black holes (BHs) between approximately 45-130M$_{\odot}$, the
so-called "pair-instability mass gap". We examine whether BHs can pollute the
gap after accreting from a stellar companion. To this end, we simulate the
evolution of isolated binaries using a population synthesis code, where we
allow for super-Eddington accretion. Under our most extreme assumptions, we
find that at most about 2% of all merging binary BH systems contains a BH with
a mass in the pair-instability mass gap, and we find that less than 0.5% of the
merging systems has a total mass larger than 90M$_{\odot}$. We find no merging
binary BH systems with a total mass exceeding 100M$_{\odot}$. We compare our
results to predictions from several dynamical pathways to pair-instability mass
gap events and discuss the distinguishable features. We conclude that the
classical isolated binary formation scenario will not significantly contribute
to the pollution of the pair-instability mass gap. The robustness of the
predicted mass gap for the isolated binary channel is promising for the
prospective of placing constraints on (i) the relative contribution of
different formation channels, (ii) the physics of the progenitors including
nuclear reaction rates, and (iii), tentatively, the Hubble parameter.
