That could be remedied by your attorney.Nobody can vouch for that more than a landlord. I have cancer and my years are numbered. One of my primary concerns is my tenants. Compared to what's around, their rent is more than reasonable and if my family sold the place, whoever buys it would increase the rents dramatically making their lives very difficult if not impossible to continue renting here. My nephew and niece swore to me they would take care of my current crop of tenants after I'm gone. I could care less what they do with the place once or if they leave. But for right now, I want to make sure they are taken care of.
I am fortunate not to be an attorney but a real estate attorney could prepare deed restrictions stating anything you wish. I've not seen it done before but I see no reason you could not state that no one shall raise the rent on such and such property for M/M John Doe more than X% per year for one year, five years, or whatever. You don't want to make it too restrictive or you'll destroy the value. The value of a rental property is in direct proportion to the rent received.
If it isn't in writing to survive you and on the record, after you croak, it's whatever the heirs want to do.