Existence of a top element for every module AEC lattice

Determine whether the lattice \(\mathscr{L}_{R}\) of abstract elementary classes of \(R\)-modules has a top element for every ring \(R\), and, if not, characterize the rings for which \(\mathscr{L}_{R}\) has a top element.

Background

The paper defines $\mathscr{L}_{R}$ as the lattice of strong submodel relations on all $R$-modules that yield an abstract elementary class and refine the direct-summand relation. It proves that when the direct-summand relation itself belongs to $\mathscr{L}_{R}$—equivalently, when $R$ is left pure semisimple—the lattice has a top element.

The unresolved issue is whether every ring admits some strongest relation in $\mathscr{L}_{R}$. If not, the authors ask for an algebraic characterization of precisely those rings whose lattices possess a top element.