Local Rupert property of the dual of the Ruperthedron

Prove that the dual of the Ruperthedron is locally Rupert (for any reasonable definition of duality invoked by the authors). Establishing this would provide a concrete counterexample candidate related to duality phenomena for Rupert’s property.

Background

The authors construct a polyhedron (the Ruperthedron) that is Rupert but not locally Rupert, and they suggest its dual as a natural candidate relevant to duality conjectures. They indicate their belief that this dual is locally Rupert but do not provide a proof.

A proof would illuminate the relationship between duality and local Rupert behavior, a theme that has surfaced repeatedly in prior work.