Whether real edge weights add strictly induced consistent path systems

Determine whether every consistent path system that is simply induced by real-valued edge weights, including possibly non-positive weights, can also be induced by positive edge weights.

Background

The paper extends the notion of simple induction to real-valued edge weights, allowing negative and zero weights, while requiring each selected path to be the unique simple geodesic. Non-positive edges in such a realization are necessarily persistent, but real weights can produce path systems that are not consistent.

The unresolved issue is whether real weights genuinely enlarge the class of consistent path systems that can be induced. The authors report that every simply induced consistent path system they have examined can also be induced by non-negative weights, but leave open whether every such system can in fact be induced by positive weights.