No right exists to demand arbitration against the director until the conditions specified in this section have been complied with. The contractor shall give the director notice in writing that the contractor claims the contract has been or will be performed fully on a day stated, which may not be less than ten days after the giving of such notice. At the time stated in the notice the director shall cause the work to be inspected, and if the director claims the work has not been completed, the director, with all reasonable dispatch, having regard to the early completion of the work, shall specify the particulars in which it is incomplete and shall direct that it be completed accordingly, or if the director considers further work necessary to bring the project up to the desired standard for acceptance either by the director or the federal highway administration, even though the director considers such contract complete, the director likewise may specify any such additional work. The contractor shall proceed with all reasonable dispatch, having due regard to weather conditions, with the performance of all such additional work with a view to a speedy completion of the project. When the contractor claims in good faith, supported by affidavit furnished to the director, that the contractor has completed such additional work according to the specifications furnished to the contractor, and the director fails for ten days to accept such work as completed, the contractor has the right to institute proceedings under this chapter.