We show that there is a non-trivial differential $d_4(h_6^3) =h_0^3g_4$ in the mod $2$ Adams spectral sequence for spheres.
This together with the results in \cite{barratt_differentials_1970,lin_differential_1998,kan_differential_2001}
completely settle the differentials of $h_i^3$ for $i\ge4$.
(The differentials of $h_i^3$ for $i=0,1,2,3$ are well-known.)
Our proof uses the Kevaire invariant elements $\theta_i \in\pi_{2^{i+1}-2}^S$ for $i=4,5$
with the properties $2\theta_4 =0$, $2\theta_5 =0$.

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