  A longstanding conjecture of Seymour states that in every oriented graph
there is a vertex whose second outneighbourhood is at least as large as its
outneighbourhood. In this short note we show that, for any fixed $p\in[0,1/2)$,
a.a.s. every orientation of $G(n,p)$ satisfies Seymour's conjecture (as well as
a related conjecture of Sullivan). This improves on a recent result of Botler,
Moura and Naia. Moreover, we show that $p=1/2$ is a natural barrier for this
problem, in the following sense: for any fixed $p\in(1/2,1)$, Seymour's
conjecture is actually equivalent to saying that, with probability bounded away
from $0$, every orientation of $G(n,p)$ satisfies Seymour's conjecture. This
provides a first reduction of the problem.
  For a second reduction, we consider minimum degrees and show that, if
Seymour's conjecture is false, then there must exist arbitrarily large
strongly-connected counterexamples with bounded minimum outdegree. Contrasting
this, we show that vertex-minimal counterexamples must have large minimum
outdegree.
