  We prove scaling limit results for the finite-volume version of the inventory
accumulation model of Sheffield (2011), which encodes a random planar map
decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn
(FK) model. In particular, we prove that the random walk associated with the
finite-volume version of this model converges in the scaling limit to a
correlated Brownian motion $\dot Z$ conditioned to stay in the first quadrant
for two units of time and satisfy $\dot Z(2) = 0$. We also show that the times
which describe complementary connected components of FK loops in the discrete
model converge to the $\pi/2$-cone times of $\dot Z$. Combined with recent
results of Duplantier, Miller, and Sheffield, our results imply that many
interesting functionals of the FK loops on a finite-volume FK planar map (e.g.
their boundary lengths and areas) converge in the scaling limit to the
corresponding "quantum" functionals of the CLE$_\kappa$ loops on a
$4/\sqrt\kappa$-Liouville quantum gravity sphere for $\kappa \in (4,8)$. Our
results are finite-volume analogues of the scaling limit theorems for the
infinite-volume version of the inventory accumulation model proven by Sheffield
(2011) and Gwynne, Mao, and Sun (2015).
