Whenever any such tide, state, school, granted, or other lands situated within the limits of any assessing district, has been included within any local improvement district by such assessing district, and the contract, leasehold, or other interest of any individual has been sold to satisfy the lien of such assessment for local improvement, the purchaser of such interest at such sale shall be entitled to receive from the state of Washington, on demand, an assignment of the contract, leasehold, or other interest purchased by him or her, and shall assume, subject to the terms and conditions of the contract or lease, the payment to the state of the amount of the balance which his or her predecessor in interest was obligated to pay.