  The LHC heavy-ion program with $^{208}$Pb$^{82+}$ beams will benefit from a
significant increase of the beam intensity when entering its High-Luminosity
era in Run~3 (2023). The stored energy is expected to surpass 20~MJ per beam.
The LHC is equipped with a betatron collimation system, which intercepts the
transverse beam halo and protects sensitive equipment such as superconducting
magnets against beam losses. However, nuclear fragmentation and electromagnetic
dissociation of $^{208}$Pb$^{82+}$ ions in collimators generates a flux of
secondary fragments, which are lost in downstream dispersion suppressor and arc
cells. These secondary ions may pose a performance limitation in upcoming runs
since they can induce magnet quenches. In order to mitigate this risk, an
alternative collimation technique, relying on bent crystals as primary
collimators, will be used in forthcoming heavy-ion runs. In this paper, we
study the power deposition in superconducting magnets by means of tracking and
\textsc{FLUKA} shower simulations, comparing the standard collimation system
against the crystal-based one. In order to quantify the predictive ability of
the simulation model, we present absolute benchmarks against beam loss monitor
measurements from the 2018 $^{208}$Pb$^{82+}$ run at 6.37~$Z$TeV. The
benchmarks cover several hundred meters of beamline, from the primary
collimators to the first arc cells. Based on these studies, we provide a
detailed analysis of ion fragmentation and leakage to the cold magnets and
quantify the expected quench margin in future $^{208}$Pb$^{82+}$ runs.
