  The purpose of this article is threefold. First, we show that when one
explores a conformal loop ensemble of parameter $\kappa=4$ ($\mathrm{CLE}_4$)
on an independent $2$-Liouville quantum gravity ($2$-LQG) disk, the surfaces
which are cut out are independent quantum disks. To achieve this, we rely on
approximations of the explorations of a $\mathrm{CLE}_4$: we first approximate
the $\mathrm{SLE}_4^{\langle \mu \rangle}(-2)$ explorations for $\mu \in
\mathbb{R}$ using explorations of the $\mathrm{CLE}_\kappa$ as $\kappa \uparrow
4$ and then we approximate the uniform exploration by letting $\mu \to \infty$.
Second, we describe the relation between the so-called natural quantum distance
and the conformally invariant distance to the boundary introduced by Werner and
Wu. Third, we establish the scaling limit of the distances from the boundary to
the large faces of $3/2$-stable maps and relate the limit to the
$\mathrm{CLE}_4$-decorated $2$-LQG.
