  Explaining the origin of supermassive black holes via a primordial origin is
severely challenged by the tight spectral distortion constraints on the
amplitude of the primordial perturbations. Following the first calculation of
how the $\mu$ constraints are modified by non-Gaussianity in a companion paper,
we here make the first robust constraints on primordial black hole formation
under large non-Gaussianity. Even the infinite $f_{\rm NL}$ limit is
insufficiently non-Gaussian but much higher-order non-Gaussianity of the form
${\cal R}={\cal R}_{\rm G}^5$ may allow the formation of any mass primordial
black hole without conflicting with distortion constraints. We caution that
such extreme models face other challenges.
