  We argue that with an increase of the collision energy, elastic
photoproduction of $\rho$ mesons on nuclei becomes affected by the significant
cross section of photon inelastic diffraction into large masses, which results
in the sizable inelastic nuclear shadowing correction to $\sigma_{\gamma A \to
\rho A}$ and the reduced effective $\rho$-nucleon cross section. We take these
effects into account by combining the vector meson dominance model, which we
upgrade to include the contribution of high-mass fluctuations of the photon
according to QCD constraints, and the Gribov-Glauber approximation for nuclear
shadowing, where the inelastic nuclear shadowing is included by means of cross
section fluctuations. The resulting approach allows us to successfully describe
the data on elastic $\rho$ photoproduction on nuclei in heavy ion UPCs in the
$7 \ {\rm GeV} < W_{\gamma p} < 46$ GeV energy range and to predict the value
of the cross section of coherent $\rho$ photoproduction in Pb-Pb UPCs at
$\sqrt{s_{NN}}=5.02$ TeV in Run 2 at the LHC, $d\sigma_{Pb Pb \to \rho Pb Pb}
(y=0)/dy= 560 \pm 25$ mb.
