Hähnle's normal-complex diameter conjecture

Let $d$ and $n$ be positive integers. A normal pure $(d-1)$-complex is a pure simplicial complex in which the link of every face of codimension at least two is connected. Its diameter is the maximum graph distance between facets in its facet-adjacency graph. Hähnle's conjecture. The diameter of every normal pure $(d-1)$-complex with $n$ vertices is bounded above by $dn$. This conjecture generalizes the polynomial Hirsch conjecture to normal pure complexes and would imply that conjecture with $k=2$ and $c=1$. It arose from the polymath3 project and remains open according to the supplied source context.