  The volume $\mathscr{B}_{\Sigma}^{{\rm comb}}(\mathbb{G})$ of the unit ball
-- with respect to the combinatorial length function $\ell_{\mathbb{G}}$ -- of
the space of measured foliations on a stable bordered surface $\Sigma$ appears
as the prefactor of the polynomial growth of the number of multicurves on
$\Sigma$. We find the range of $s \in \mathbb{R}$ for which
$(\mathscr{B}_{\Sigma}^{{\rm comb}})^{s}$, as a function over the combinatorial
moduli spaces, is integrable with respect to the Kontsevich measure. The
results depends on the topology of $\Sigma$, in contrast with the situation for
hyperbolic surfaces where Arana-Herrera and Athreya (arXiv:1907.06287) recently
proved an optimal square-integrability.
