Hähnle's linear-in-dimension 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 conjecture. The diameter of every $d$-polytope with $n$ facets is bounded above by $dn$. This is presented as a more explicit version of the polynomial Hirsch conjecture; the source says that the underlying diameter problem remains largely open and attributes this formulation to N. Hähnle.