Conforti–Johnson's Petersen-minor T-join min-max conjecture

Let $G$ be a graph, and let $T$ be the set of its odd-degree vertices. A $T$-join is a subgraph whose odd-degree vertices are precisely the vertices in $T$, and a $T$-cut is a cut $(A,B)$ such that $|T\cap A|$ is odd. Conforti–Johnson's Petersen-minor T-join min-max conjecture. If $G$ has no Petersen minor, then the maximum number of edge-disjoint $T$-joins equals the size of the smallest $T$-cut. The paper states that this conjecture would follow from Seymour's Petersen-minor edge-coloring conjecture; its general resolution is not given here.