Upon the earlier of the resignation of a late term appointee or the conclusion of the sixtieth day of the term of the succeeding Governor, that appointed office shall be considered vacant. The succeeding Governor may then make an appointment to fill that vacancy, regardless of whether the statute that creates the appointed office provides for appointment to fill a vacancy. All other requirements of law applicable to that appointed office shall apply to the succeeding Governor’s appointee, including but not limited to eligibility, qualifications, and confirmation by the Senate.