Discrete homotopy of graphs and $k$-parabolic subspace arrangements

Discrete homotopy theory is a combinatorial homotopy theory defined for discrete objects. Barcelo–Severs–White conjectured that the homotopy groups of complements of $k$-parabolic subspace arrangements are isomorphic to the discrete homotopy groups of graphs associated with the Coxeter complexes. We show that this correspondence fails in general by giving a counterexample in degree two for the $4$-parabolic subspace arrangement of type $A_5$.