Finite generation of graph configuration-space homology as a topological graph-injection module

For a graph $G$, let $F_n(G)$ be the configuration space of $n$ ordered points in $G$, and let $cmathcal{TGI}$ be the category of topological graph embeddings. The assignment $G\mapsto H_i(F_n(G))$ defines a $cmathcal{TGI}$-module. Universal finite generation conjecture. For any natural numbers $i$ and $n$, the $cmathcal{TGI}$-module $H_i(F_n(\bullet))$ is finitely generated. This is a categorical reformulation of universal finite generation for all graphs. It is supported by known cases, including $i=1$ for all graphs and several restricted graph classes, but remains open in general.