  Using the Gribov-Glauber model for photon-nucleus scattering and a
generalization of the vector meson dominance model for the hadronic structure
of the photon, we make predictions for the cross section of incoherent $\rho$
photoproduction in Pb-Pb ultraperipheral collisions (UPCs) in the Large Hadron
Collider kinematics. We find that the effect of the inelastic nuclear shadowing
is significant and leads to an additional 25% suppression of the incoherent
cross section. Comparing our predictions to those of the STARlight Monte Carlo
framework, we observe very significant differences.
