Directed-graph torsion in magnitude (co)homology beyond the image of the bidirection-forgetful functor

Establish whether there exist directed graphs whose magnitude (co)homology exhibits torsion not arising from any undirected graph via the functor ρ: Graph → Quiver that makes each undirected edge bidirectional; specifically, construct a directed graph D such that torsion in MH_*^*(D) does not coincide with torsion from MH_*^*(ρ(G)) for any undirected graph G, or prove that all torsion in magnitude (co)homology of digraphs lies in the essential image of ρ.

Background

Magnitude homology for quivers extends the undirected-graph theory. Prior works have exhibited torsion in magnitude (co)homology of graphs, and this paper proves finite generation results for quiver magnitude cohomology in bounded-genus settings.

The authors explicitly state that it is unknown whether torsion observed for quivers always comes from undirected graphs via the functor ρ that forgets orientation by making edges bidirectional, and pose a concrete question about constructing genuinely directed torsion phenomena.