The even-cycle decomposition conjecture for odd-$K_5$-minor-free signed graphs

A signed graph is a graph whose edges are designated even or odd; a cycle is even when it contains an even number of odd edges. A graph is even cycle decomposable if its edge set can be partitioned into even cycles. The graph is Eulerian if every vertex has even degree, and odd-$K_5$-minor free if it has no odd minor of $K_5$. Even-cycle decomposition conjecture. Every $2$-connected Eulerian loopless odd-$K_5$-minor-free signed graph with an even number of odd edges is even cycle decomposable. This conjecture extends the known results for bipartite graphs and graphs with no $K_5$-minor, where the corresponding necessary conditions are sufficient for an even-cycle decomposition.