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, $dsigma_{Pb Pb to rho Pb Pb} (y=0)/dy= 560 pm 25$ mb.
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