Whenever the board of commissioners of any drainage district shall have heretofore issued any warrants of the district for the purpose of completing a system of drainage for such district so as to produce the benefits to the lands of the district found by the jury to be benefited by the proposed improvement as provided in the preceding section, and the total estimated maximum benefits found by the jury that would accrue to the lands of the district by reason of such proposed improvement are not sufficient to cover the actual cost of such improvement, including the cost of completing the same as hereinabove provided, the board of commissioners of such district shall file a petition in the superior court in the original proceeding for the determination of the damages and benefits to accrue from the proposed improvement, setting forth the facts, describing the lands that have been, in the judgment of the commissioners, actually benefited by the completed improvement, stating the estimated amount of benefits per acre that have accrued to each tract of land respectively, giving the name of the owner or reputed owner of such tract of land, and praying that the original proceedings be opened for further proceedings for the purpose of determining the benefits which have accrued to each tract of land actually benefited by the completed improvement. If the said board of commissioners fail or refuse to file such petition within sixty days after receipt of a written request so to do, signed by any warrant-holder, then the said warrant-holder shall have the right to file same.