Vertex-primitive s-arc-transitive digraphs of symplectic groups

A digraph is s-arc-transitive if its automorphism group is transitive on directed paths with s edges, that is, on s-arcs. Although infinite families of finite s-arc transitive digraphs of arbitrary valency were constructed by the third author in 1989, existence of a vertex-primitive 2-arc-transitive digraph was not known until an infinite family was constructed by the second author with Li and Xia in 2017. This led to a conjecture by the second author and Xia in 2018 that, for a finite vertex-primitive s-arc-transitive digraph, s is at most 2, together with their proof that it is sufficient to prove the conjecture for digraphs with an almost simple group of automorphisms. This paper confirms the conjecture for finite symplectic groups.