This happened to a rental beside me (the old tenants had just bought their own home. New tenants are paying 900 euros a month more than the previous tenant.I know someone who sold a house that was rented out well below market rent. To the amount of between €550-€600 pcm.
He sold up anyway and the estate agent did tell him that when people rang up enquiring about the property, once they heard the rent (it was in a RPZ) they weren't interested. If was purchased as a buy to let again and the rent was quoted on daft as the market rent. That is, the new owners just upped the rent by €550-€600 pcm. The property was renter in three days. RTB are beyond useless. There's no enforcement. My advice is just raise the rent. The only people who will know are the old tenants, who will be long gone.
The RPZ system only works where the existing tenant is still in place, and believe me, they will hang on as long as they can because they have no real choice given the circumstances. There is no enforcement whatsoever if the old tenant moves of their own accord, because its no longer of any interest, and the new tenant will not know what the previous rent was. This would only work if there was a publicly available register of rents that was accurate, but nobody has suggested that.