  Circuit diameters of polyhedra are a fundamental tool for studying the
complexity of circuit augmentation schemes for linear programming and for
finding lower bounds on combinatorial diameters. The main open problem in this
area is the circuit diameter conjecture, the analogue of the Hirsch conjecture
in the circuit setting. A natural question is whether the well-known
counterexamples to the Hirsch conjecture carry over. Previously, Stephen and
Yusun showed that the Klee-Walkup counterexample to the unbounded Hirsch
conjecture does not transfer to the circuit setting. Our main contribution is
to show that the original counterexamples for the other variants, for bounded
polytopes and using monotone walks, also do not transfer.
  Our results rely on new observations on structural properties of these
counterexamples. To resolve the bounded case, we exploit the geometry of
certain $2$-faces of the polytopes underlying all known bounded Hirsch
counterexamples in Santos' work. For Todd's monotone Hirsch counterexample, we
provide two alternative approaches. The first one uses sign-compatible circuit
walks, and the second one uses the observation that Todd's polytope is
anti-blocking. Along the way, we enumerate all linear programs over the
polytope and find four new orientations that contradict the monotone Hirsch
conjecture, while the remaining $7107$ satisfy the bound.
