  Consider a supercritical branching random walk on the real line. The
consistent maximal displacement is the smallest of the distances between the
trajectories followed by individuals at the $n$th generation and the boundary
of the process. Fang and Zeitouni, and Faraud, Hu and Shi proved that under
some integrability conditions, the consistent maximal displacement grows almost
surely at rate $\lambda^* n^{1/3}$ for some explicit constant $\lambda^*$. We
obtain here a necessary and sufficient condition for this asymptotic behaviour
to hold.
