Any person interested in any real estate affected by said assessment may, within the time fixed, appear and file objections. As to all parcels, lots, or blocks as to which no objections are filed, within the time as aforesaid, the assessment thereon shall be confirmed and shall be final. On the hearing, each person may offer proof, and proof may also be offered on behalf of the assessment, and the board shall affirm, modify, change, and determine the assessment, in such sum as to the board appears just and right. The commissioners may increase the assessment during such hearing upon any particular tract by mailing notice to the owner at his or her last known address, to be and appear within a time not less than ten days after the date of the notice, to show cause why his or her assessment should not be increased. When the assessment is finally equalized and fixed by the board, the secretary thereof shall certify the same to the county treasurer of each county in which the lands are situated, for collection; or if appeal has been taken from any part thereof, then so much thereof as has not been appealed from shall be certified. In case any owner of property appeals to the superior court in relation to the assessment or other matter when the amount of the assessment is determined by the court finally, either upon determination of the superior court, or review by the supreme court or the court of appeals, then the assessment as finally fixed and determined by the court shall be certified by the clerk of the proper court to the county treasurer of the county in which the lands are situated and shall be spread upon and become a part of the assessment roll hereinbefore referred to.