None of the executive officers mentioned in this article shall hold any other office during the term of his service. A person who has been elected or who has served as governor during all or any part of two consecutive terms shall be ineligible for the office of governor during any part of the term immediately following the second of the two consecutive terms. The person holding the office of governor when this section is ratified shall not be prevented from holding the office of governor during the term immediately following the term he is then serving.