Robertson's labelled conjecture for topological minors

Let $k$ be a positive integer. Let $G_1,G_2,\ldots$ be graphs that do not contain a Robertson chain of length at least $k$ as a topological minor. Let $Q$ be a set equipped with a well-quasi-ordering $\leq_Q$, and for each positive integer $i$, let $\phi_i:V(G_i)\rightarrow Q$ be a function. A homeomorphic embedding from a graph $H$ to a graph $G$ is an injective map on vertices together with paths or cycles assigned to edges, satisfying the usual disjointness and incidence conditions; write it as $\eta:H\hookrightarrow G$. Robertson's labelled conjecture. There exist integers $j,j'$ with $1\leq j<j'$ and a homeomorphic embedding $\eta:G_j\hookrightarrow G_{j'}$ such that

$$\phi_j(v)\leq_Q\phi_{j'}(\eta(v))$$

for every $v\in V(G_j)$. This is the labelled strengthening of Robertson's conjecture and is formulated using the equivalent homeomorphic-embedding characterization of topological minors; its resolution is not supplied here and remains open.