After 30 days after the date of mailing by the contractor or his assigns of the statement to the property owners pursuant to Section 5390, the contractor, or his assigns, shall sign and file in the office of the superintendent of streets a written statement of all payments received upon the assessment. Thereupon, the superintendent of streets shall file the statement so made with the record of the warrant and assessment by attaching it in the assessment book immediately following the record of the assessment. If under the proceedings being taken bonds are to be issued to represent unpaid assessments, no further payments of assessments can be accepted by the contractor or his assigns after filing said statement, as to any assessment which is to be represented by a bond. When payments are made to the superintendent of streets pursuant to Section 5396 of this division, it shall not be necessary for the contractor or his assigns to file said written statement of payments received.