The homotopy-test-graph conjecture for connected bipartite graphs

A graph is a homotopy test graph if its Hom-complexes detect graph colorability through the corresponding homotopy-connectivity bound. Homotopy-test-graph conjecture. Every connected bipartite graph is a homotopy test graph. The claim generalizes the known case of connected bipartite graphs equipped with a ${\mathbb Z}_2$-action that flips an edge; the source gives no resolution of the general statement.