If a vacancy occurs in the office of governor and the regularly elected lieutenant governor succeeds to the office of governor as provided by art. III, § 11, of the constitution, or if the office of lieutenant governor otherwise becomes vacant, the person designated as next successor to the office of lieutenant governor as provided in AS 44.19.040 succeeds to the office of lieutenant governor for the remainder of the term vacated, or until a special election is held. When the person appointed under AS 44.19.040 succeeds to the office of lieutenant governor, the governor shall appoint a person to succeed to the office of lieutenant governor in case of a subsequent vacancy.