Can they waive it? Yes they can. Will they waive it? In my experience, unlikely. A landlord has a legitimate right to assure itself that the assignee is capable of meeting the requirements of the lease - and that includes being able to pay the rent.
Let's leave the ethical/compassionate aspect aside - there's an argument there that allowing someone to take over a lease which would ultimately fail and cost them more money in the long run is in itself unethical.
Ok thank you - they have however already viewed references and bank statements.