When the board, or landowners in districts having no board, certify under oath to the board of supervisors and show to its satisfaction that the reclamation works are completed, or that two dollars ($2) per acre has been expended for reclamation works on swamp or overflowed lands purchased pursuant to an application filed with the State Lands Commission prior to September 19, 1939, the board of supervisors shall certify such facts to the State Lands Commission.