  We pursue the analysis of nesting statistics in the $O(n)$ loop model on
random maps, initiated for maps with the topology of disks and cylinders in
math-ph/1605.02239, here for arbitrary topologies. For this purpose we rely on
the topological recursion results of math-ph/0910.5896 and math-ph/1303.5808
for the enumeration of maps in the $O(n)$ model. We characterize the generating
series of maps of genus $g$ with $k'$ marked points and $k$ boundaries and
realizing a fixed nesting graph. These generating series are amenable to
explicit computations in the loop model with bending energy on triangulations,
and we characterize their behavior at criticality in the dense and in the
dilute phase.
