Degreewise Cut-Semigroup Saturation for $K_5$-Minor-Free Graphs and Seymour's Planar Edge-Colouring Conjecture

For every positive integer $k$, we prove that the homogeneous cut semigroup of every $K_5$-minor-free graph is saturated at height $k$ if and only if every planar $k$-graph is $k$-edge-colourable. Here a $k$-graph is a loopless $k$-regular multigraph in which every odd vertex cut has size at least $