  We study the volume of rigid loop-$O(n)$ quadrangulations with a boundary of
length $2p$ in the critical non-generic regime. We prove that, as the
half-perimeter $p$ goes to infinity, the volume scales in distribution to an
explicit random variable. This limiting random variable is described in terms
of the multiplicative cascades of Chen, Curien and Maillard arXiv:1702.06916,
or alternatively (in the dilute case) as the law of the area of a unit-boundary
$\gamma$-quantum disc, as determined by Ang and Gwynne arXiv:1903.09120, for
suitable $\gamma$. Our arguments go through a classification of the map into
several regions, where we rule out the contribution of bad regions to be left
with a tractable portion of the map. One key observable for this classification
is a Markov chain which explores the nested loops around a size-biased vertex
pick in the map, making explicit the spinal structure of the discrete
multiplicative cascade.
