All-size directed-cut bound

Prove that for every integer d≥2 and every digraph G with m edges and minimum outdegree at least d, there exists a vertex bipartition V(G)=A∪B such that both directed cuts satisfy e_G(A,B)≥⌈(d−1)m/[2(2d−1)]⌉ and e_G(B,A)≥⌈(d−1)m/[2(2d−1)]⌉, without requiring m to be sufficiently large.

Background

The paper proves the stronger additive bound with an additional term d²/[2(2d−1)] for all sufficiently large m, and proves the bound ⌈m/6⌉ for every m when d=2. The authors note that the large-m theorem can fail for small m, but conjecture that the coefficient-only bound c_d m, with c_d=(d−1)/[2(2d−1)], remains achievable for every m. This is identified as the remaining open part of Scott’s question for general d.