The Monotonic Bounded Hirsch Conjecture is False for Dimension at Least 4

We exhibit a d-polytope P with n facets and a linear form ϕ such that all paths from a certain vertex of P to a ϕ-minimizing vertex that are nonincreasing in ϕ have length at least n − d + 1, provided d ≥ 4 and n − d ≥ 4. Indeed the discrepancy between the minimum length of such paths and n − d is at least min{[d/4], [(n − d)/4]} for some such polytope.