Hähnle's sharpened linear Hirsch conjecture

Let $P$ be a $d$-dimensional polytope with $n$ facets, and let its diameter be the maximum length of a shortest edge path between two vertices of $P$. Hähnle's sharpened conjecture. The diameter of every $d$-polytope with $n$ facets is bounded above by $d(n-d)$. The source states that, using an earlier remark, this formulation is almost equivalent to the bound $dn$; the question of general diameter bounds remains open.