Packing odd T-joins with at most two terminals

Take a graph G, an edge subset Σ ⊆ E(G), and a set of terminals T ⊆V(G) where |T| is even. The triple (G,Σ,T) is called a signed graft. A T-join is odd if it contains an odd number of edges from Σ. Let ν be the maximum number of edge-disjoint odd T-joins. A signature is a set of the form Σ△δ(U) where U ⊆V(G) and |U∩T| is even. Let τ be the minimum cardinality a T-cut or a signature can achieve. Then v≤ τ and we say that (G, Σ, T) packs if equality holds here. We prove that (G, Σ, T) packs if the signed graft is Eulerian and it excludes two special nonpacking minors. Our result confirms the Cycling Conjecture for the class of clutters of odd T-joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two-commodity paths, packing T-joins with at most four terminals, and a new result on covering edges with cuts.