Robertson's conjecture and universal finite generation in the homology of graph braid groups

We formulate a categorification of Robertson’s conjecture analogous to the categorical graph minor conjecture of Miyata–Proudfood–Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs – in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Miyata–Proudfoot, implying that the category of cographs is Noetherian, a result of potential independent interest.