The board of directors, to whom such petition is presented, may require as a condition precedent to the granting of the same, that the petitioners shall severally pay to such district such respective sums, as nearly as the same can be estimated by the board, as said petitioners or their grantors would have been required to pay to such district as assessment for the payment of their pro rata share of all bonds and interest thereon, which may have previously thereto been issued by said district, or for the payment of the pro rata share of the cost of construction under any contract between the district and the United States accompanying which bonds of the district have not been deposited with the United States as in this act provided, had such lands been included in such district at the time the same was originally formed or when said bonds were so issued, or when said contract with the United States was made; provided, that in case unentered public land is proposed to be annexed to the district the board of directors of the district instead of requiring such payment as a condition precedent, may assess such charges against such unentered public land upon the records of the district to be collected in the manner authorized by the said act of congress of August 11, 1916.