Non-well-quasi-ordering of k-connected bipartite graphs by bipartite minors

Let $k$ be a positive integer, and consider the class of $k$-connected bipartite graphs with the bipartite minor relation. Bipartite-minor non-well-quasi-order conjecture. There exists no $k$ such that the class of $k$-connected bipartite graphs is well-quasi-ordered by the bipartite minor relation. This conjecture extends the proved failure of well-quasi-ordering for $2$-connected bipartite graphs, asserting that increasing the connectivity requirement never yields a well-quasi-order.