Categorical Robertson conjecture

For $k>0$, let $cmathcal{GTM}_k$ be the full subcategory of topological minor morphisms spanned by graphs not admitting the Robertson chain $R_k$ as a topological minor, and let $Q$ be a well-quasi-order used to label vertices. Write $cmathcal{GTM}_{k,Q}$ for the corresponding category of $Q$-labeled graphs. Categorical Robertson conjecture. Fix $k>0$ and a well-quasi-order $Q$. The category $cmathcal{GTM}_{k,Q}$ is Noetherian over any Noetherian ring. This categorifies Robertson's conjecture and, together with the categorical graph minor conjecture, is intended to imply universal finite generation for graph braid-group homology. The case $k=1$ is proved by work of Barter and Proudfoot–Ramos; the general conjecture remains open.