Whenever in the progress of the construction of the system of improvement it shall become necessary to construct a portion of such system across any public or other road or public utility, the board of supervisors, or in case the work is being done by contract the board of county commissioners, shall serve notice in writing upon the public officers, corporation, or person having charge of, or controlling or owning such road or public utility, as the case may be, of the present necessity of such crossing, giving the location, kind, dimensions, and requirement thereof, for the purpose of the system of improvement, and stating a reasonable time, to be fixed by the county engineer, within which plans for such crossing must be filed for approval in case the public officers, corporation, or person controlling or owning such road or public utility desire to construct such crossing. As soon as convenient, within the time fixed in the notice, the public officers, corporation, or person shall, if they desire to construct such crossing, prepare and submit to the county engineer for approval duplicate detailed plans and specifications for such crossing. Upon submission of such plans, the county engineer shall examine and may modify the same to meet the requirements of the system of improvement, and when such plans or modified plans are satisfactory to the county engineer, he or she shall approve the same and return one thereof to the public officers, corporation, or person submitting the same, and file the duplicate in his or her office, and shall notify such public officers, corporation, or person of the time within which said crossing must be constructed. Upon the return of such approved plans, the public officers, corporation, or person controlling such road or public utility shall, within the time fixed by the county engineer, construct such crossing in accordance with the approved plans, and shall thereafter maintain the same. In case such public officers, corporation, or person controlling or owning such road or public utility shall fail to file plans for such crossing within the time prescribed in the notice, the board of supervisors or of county commissioners, as the case may be, shall proceed with the construction of such crossing in such manner as will cause no unnecessary injury to or interference with such road or public utility. The cost of construction and maintenance of only such crossings or such portion of such cost as would not have been necessary but for the construction of the system of improvement shall be a proper charge against the improvement district, and only so much of such cost as the board of county commissioners shall deem reasonable shall be allowed as a charge against the district in the case of crossings constructed by others than the district. The amount of costs of construction allowed as a charge against the district by the board of county commissioners shall be credited on the assessments against the property on which the crossing is constructed, and any excess over such assessment shall be paid out of the funds of the district.