  Determining an upper bound on $s$ for finite vertex-primitive
$s$-arc-transitive digraphs has received considerable attention dating back to
a question of Praeger in 1990. It was shown by Giudici and Xia that the
smallest upper bound on $s$ is attained for some digraph admitting an almost
simple $s$-arc-transitive group. In this paper, based on the work of Pan, Wu
and Yin, we prove that $s\leqslant 2$ in the case where the group is an
alternating or symmetric group.
