Unless the Joint Legislative Budget Committee within 60 days after receipt of such notice advises the board in writing that the Legislature desires to reconsider any part of the withheld portion, the board shall institute proceedings to compel the Controller to issue his warrant for the balance of the omnibus claim appropriation. The board shall prosecute any such proceeding to final determination and shall disburse the funds finally paid over to it. Funds subject to such proceedings shall continue to be available until the end of the fiscal year in which final determination of the proceeding occurs.