Hirsch bound conjecture for the circuit diameter of polyhedra

Let $P$ be an $n$-dimensional polyhedron with $f$ facets. Its circuit diameter is the maximum number of maximum-length steps along circuit directions needed to go from one vertex of $P$ to another.

Circuit diameter bound. For any $n$-dimensional polyhedron with $f$ facets, the circuit diameter is bounded above by

$$f-n.$$

The circuit diameter is an analogue of the combinatorial diameter based on circuit directions and gives a lower bound on the number of augmentation steps under selection rules for linear programming. The analogous Hirsch bound for combinatorial diameter is false in general, so whether it always holds for circuit diameter is the question posed here.