Strong orientation of a connected graph for a crossing family

Given a connected graph G = (V,E) and a crossing family C over ground set V such that |δG(U)| ≥ 2 for every U ∈ C, we prove there exists a strong orientation of G for C, i.e., an orientation of G such that each set in C has at least one outgoing and at least one incoming arc. This implies the main conjecture in Chudnovsky et al. (Disjoint dijoins. Journal of Combinatorial Theory, Series B, 120:18–35, 2016).