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 $λ^* n^{1/3}$ for some explicit constant $λ^*$. We obtain here a necessary and sufficient condition for this asymptotic behaviour to hold.