Robertson's conjecture on bounded Robertson chains

Let a Robertson chain of length $k$ be the graph obtained from a path of length $k$ by duplicating each edge. A graph contains another graph as a topological minor if the latter can be obtained from a subgraph of the former by repeatedly contracting edges incident with vertices of degree two. Robertson's conjecture. For every positive integer $k$, graphs that do not contain the Robertson chain of length $k$ as a topological minor are well-quasi-ordered by the topological minor relation. This conjecture identifies the known obstruction to well-quasi-ordering graphs under topological containment and is the main objective of the paper; the bounded tree-width case is proved here, while the general conjecture remains open.