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I've had no problem with Water and even Windex in removing old foam rubber padding and rubber egg-crate padding. Just saturate the padding, let soak for 15-minutes and scrape clean off with a putty knife (metal or plastic).
Scrape lightly to avoid damaging the floor and scrape down to what's stuck. Then, you can re-apply and re-soak, with Water or Windex, the more stubborn areas until fully removed. Only the rubber egg-crate padding ever left permanent light marks in the floor beneath.