  Sheffield (2011) introduced an inventory accumulation model which encodes a
random planar map decorated by a collection of loops sampled from the critical
Fortuin-Kasteleyn (FK) model. He showed that a certain two-dimensional random
walk associated with an infinite-volume version of the model converges in the
scaling limit to a correlated planar Brownian motion. We improve on this
scaling limit result by showing that the times corresponding to FK loops (or
"flexible orders") in the inventory accumulation model converge in the scaling
limit to the $\pi/2$-cone times of the correlated Brownian motion. This
statement implies a scaling limit result for the joint law of the areas and
boundary lengths of the bounded complementary connected components of the FK
loops on the infinite-volume planar map. In light of the encoding of
Duplantier, Miller, and Sheffield (2014), the limiting object coincides with
the joint law of the areas and boundary lengths of the bounded complementary
connected components of a collection of CLE$_\kappa$ loops on an independent
Liouville quantum gravity cone.
