C. The Board shall furnish a statement of the several shares of the cost so determined to the governing body of each county, city and town involved, and the respective shares shall be paid within thirty days from the date upon which such statement is furnished. If the costs of any such repairs, improvements or additions will not exceed $2000 they may be authorized by the governing body of the county, city or town to whose correctional facility such repairs, improvements or additions are to be made. If the costs will exceed $2000, such repairs, improvements or additions shall be recommended by the Board and agreed on in advance by the governing bodies of both or all of the counties, cities and towns involved. In case of disagreement, the matter of the extent of the repairs, improvements or additions and the proportionate cost to the respective localities involved shall be determined by the circuit court of the locality which owns or maintains the correctional institution proposed to be repaired, improved or added to, upon the petition of the Board.