Capping rental rates to mortgages is the best solution. Whatever the mortgage is, you can't charge a penny more than that.
Any tax burden will be passed to the renter without a cap on rental rates.
A use it or lose it policy would also be effective. Rent or sell an unoccupied house or it is turned over to the city council at half tax assessment value.
You need to have teeth, actual losses to encourage them to move.
Oh man, if my batteries run out on my noise cancelling headphones I leave them on so I don't have to talk to people. Now I can enjoy the, "I know those aren't on" line.