Relationship between regularity and universal presentations

Determine the relationship between the regularities of the homology modules \(H_\bullet(B\Gamma)\) over \(F[E]\) and finite universal presentations by finite sets of atomic graphs supplying generators and relations for homology groups of graphs.

Background

The paper discusses universal presentations consisting of finite sets of atomic graphs that would provide generators and relations for $H_i(B\Gamma)$ uniformly over all graphs $\Gamma$. Since arbitrarily large regularity could require generators or relations involving arbitrarily many particles, the existence of finite universal presentations may fail. The authors explicitly ask how these two phenomena are related.