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We show, assuming the Strong Exponential Time Hypothesis, that for every$\varepsilon > 0$, approximating directed Diameter on $m$-arc graphs withinratio $7/4 - \varepsilon$ requires $m^{4/3 - o(1)}$ time. Our construction usesnonnegative edge weights but even holds for sparse digraphs, i.e., for whichthe number of vertices $n$ and the number of arcs $m$ satisfy $m = n\log^{O(1)} n$. This is the first result that conditionally rules out anear-linear time $5/3$-approximation for Diameter.

Inapproximability of Diameter in super-linear time: Beyond the 5/3 ratio