  From the point of view of optimization, a critical issue is relating the
combinatorial diameter of a polyhedron to its number of facets $f$ and
dimension $d$. In the seminal paper of Klee and Walkup [KW67], the Hirsch
conjecture of an upper bound of $f-d$ was shown to be equivalent to several
seemingly simpler statements, and was disproved for unbounded polyhedra through
the construction of a particular 4-dimensional polyhedron $U_4$ with 8 facets.
The Hirsch bound for bounded polyhedra was only recently disproved by Santos.
  We consider analogous properties for a variant of the combinatorial diameter
called the circuit diameter. In this variant, the walks are built from the
circuit directions of the polyhedron, which are the minimal non-trivial
solutions to the system defining the polyhedron.
  We are able to prove that circuit variants of the so-called non-revisiting
conjecture and $d$-step conjecture both imply the circuit analogue of the
Hirsch conjecture. For the equivalences in [KW67], the wedge construction was a
fundamental proof technique. We exhibit why it is not available in the circuit
setting, and what are the implications of losing it as a tool.
  Further, we show the circuit analogue of the non-revisiting conjecture
implies a linear bound on the circuit diameter of all unbounded polyhedra - in
contrast to what is known for the combinatorial diameter. Finally, we give two
proofs of a circuit version of the $4$-step conjecture. These results offer
some hope that the circuit version of the Hirsch conjecture may hold in
general.
  A challenge in the circuit setting is that different realizations of
polyhedra of the same combinatorial structure may have different diameters. We
adapt the notion of simplicity to work with circuits in the form of C-simple
and wedge-simple polyhedra. We show that it suffices to consider such
polyhedra.
