If a district elects pursuant to Section 13034 to have the district board consist of five members elected from the district, the district board may call a special election for the purpose of electing a district board member to succeed the supervisorial member on the district board. A special election called pursuant to this section shall be called and conducted in accordance with the provisions of Chapter 4 (commencing with Section 13050) of this division. A person elected to a district board at a special election pursuant to this section shall hold office until the next succeeding general district election and until the election and qualification of his successor. Thereafter, the term of office of such elected board member shall be four years.

If a special election is not called, the successor to the supervisorial member shall be elected at the next general district election.

Any supervisor serving as a district board member when an election is held pursuant to this section shall remain in office until the election and qualification of his successor.

Any district electing to call a special election pursuant to the provisions of this section shall reimburse any local government for any costs incurred by such local government by reason of the provisions of this section.