The weighted expanding formulation of Seymour's conjecture

Let $D$ be a digraph without loops or two-cycles. Give each vertex a nonnegative real weight, and call a vertex weakly expanding when the weight of its second out-neighborhood is at least the weight of its first out-neighborhood. Weighted expanding formulation of Seymour's conjecture. Every digraph without loops or two-cycles has a vertex-weighting such that every vertex is weakly expanding. By the preceding argument, this formulation is equivalent to Seymour's second neighborhood conjecture. The paper uses it to relate the conjecture to an alternative involving strongly contracting vertices in the reverse digraph; it does not resolve the general case.