Upon the adoption and filing of the plan for reclamation, the owners of a majority in acres of the lands in the drainage district if they desire to proceed with the work of reclaiming the drainage district, or the owners of a majority in acres of the lands in any subdrainage district, if they desire to proceed with the work of reclaiming the lands in the sub-drainage district, may respectively present a petition to that effect to the board of commissioners of the drainage district, signed by the landowners.  When so petitioned, the board of commissioners of the drainage district shall proceed with the reclamation of the district or sub-district, the incurring of debt, the issuance of bonds, and the levy, assessment, and collection of taxes as hereinafter provided.