On a conjecture of trietsch and handler on the flow-dependent steiner ratio

Recently Trietsch and Handler extended a conjecture of Pollak and Gilbert on the Steiner ratio to a model first studied by Gilbert where the cost of an edge is flow dependent. For three given points Trietsch and Handler showed that the maximum Steiner ratio is achieved when the cost is independent of the flow, i. e., the extended conjecture is reduced to the original conjecture. They also conjectured that the two conjectures are equivalent for any number of given points. In this paper we give a counterexample to their conjecture if the number of given points is at least four. We also replace their proof for the three-point case, which depends heavily on computer-aided calculations, by a simple geometric proof.