The Υ(2S) and Υ(3S) production cross-sections are measured relative to that of the Υ(1S) meson, as a function of charged-particle multiplicity in proton-proton collisions at a centre-of-mass energy of 13 TeV. The measurement uses data collected by the LHCb experiment in 2018 corresponding to an integrated luminosity of 2 fb−1. Both the Υ(2S)-to-Υ(1S) and Υ(3S)-to-Υ(1S) cross-section ratios are found to decrease significantly as a function of event multiplicity, with the Υ(3S)-to-Υ(1S) ratio showing a steeper decline towards high multiplicity. This hierarchy is qualitatively consistent with the comover model predictions, indicating that final-state interactions play an important role in bottomonia production in high-multiplicity events.