Strong conjecture on pre-quasi Gelfand triples

A pre-quasi Gelfand triple is a triple satisfying the pre-quasi Gelfand triple axioms, while a quasi Gelfand triple satisfies the stronger quasi Gelfand triple axioms.

Strong conjecture. Every pre-quasi Gelfand triple is a quasi Gelfand triple.

This is the stronger version of the conjecture for arbitrary pre-quasi Gelfand triples, without the Hilbert-space restriction. The authors consider it much more difficult because much of the available Hilbert-space theory does not apply; it remains open.