  In this paper we investigate pointed $(\mathbf{q}, g, n)$-Boltzmann
loop-decorated maps with loops traversing only inner triangular faces. Using
the peeling exploration of arXiv:1809.02012 modified to this setting we show
that its law in the non-generic critical phase can be coded in terms of a
random walk confined to the positive integers by a new specific boundary
condition. Under a technical assumption that we believe to be true, combining
this observation with explicit quantities for the peeling law we derive the
large deviations property for the distribution of the so-called nesting
statistic and show that the exploration process possesses exactly the same
scaling limit as in the rigid loop model on bipartite maps that is a specific
self-similar Markov process introduced in arXiv:1809.02012. Besides, we
conclude the equivalence of the admissible weight sequences related by the
so-called fixed point equation by proving the missing direction in the argument
of arXiv:1202.5521.
