  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.
