We show, assuming the Strong Exponential Time Hypothesis, that for every $varepsilon > 0$, approximating directed Diameter on $m$-arc graphs within ratio $7/4 - varepsilon$ requires $m^{4/3 - o(1)}$ time. Our construction uses nonnegative edge weights but even holds for sparse digraphs, i.e., for which the 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 a near-linear time $5/3$-approximation for Diameter.
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