If an appeal has been made in reference to the boundaries of the assessment district or to the tentative assessment as provided in section 6117.09 of the Revised Code, and it appears from the transcript certified to the board of county commissioners by the probate judge that the boundaries of the assessment district should be changed or that the tentative assessment should be modified, the board may make such changes or modifications and proceed with the improvement, provided that the necessity of such improvement has not been denied by the probate court or the cost thereof found to exceed the benefits resulting from such improvement.
R.C. § 6117.20