The $d$-step Conjecture for polytope diameters

Let $H(n,d)$ denote the maximum graph diameter of a $d$-dimensional polytope with $n$ facets.

The $d$-step Conjecture. For every $d \geq 2$,

$$H(2d,d) \leq d.$$

The source states that this conjecture is equivalent to the Hirsch Conjecture by the Klee–Walkup theorem. Since the Hirsch conjecture is refuted, this conjecture is also refuted.