Before revision:  This settles the SETH-hardness of approximating the diameter of sparse directed unweighted graphs for deterministic reductions, up to NSETH , and for randomized reductions, up to NUNSETH.
After revision:  This settles the SETH-hardness of approximating the diameter of sparse directed unweighted graphs for deterministic reductions, up to NSETH . We make the same characterization for randomized SETH-based reductions, assuming another hypothesis called NUNSETH.