It is the duty of the executor to file with the department within 60 days after a final determination of a deficiency in federal estate tax has been made, written notice of the deficiency. If, based upon this deficiency and the ground for it, it appears that the amount of tax previously paid is less than the amount of tax owing, the difference together with interest at the rate of seven percent a year from the due date of the tax shall be paid upon notice and demand by the department. If the executor fails to give the notice required by this section, any additional tax owing may be assessed, or a proceeding in court for the collection of the tax may be begun without assessment at any time before the filing of notice or within 30 days after the delinquent filing of notice, notwithstanding the provisions of AS 43.05.260.