Weak conjecture on pre-quasi Gelfand triples of Hilbert spaces

A pre-quasi Gelfand triple of Hilbert spaces is a pre-quasi Gelfand triple whose underlying spaces are Hilbert spaces. A quasi Gelfand triple is a triple satisfying the quasi Gelfand triple axioms.

Weak conjecture. Every pre-quasi Gelfand triple of Hilbert spaces is a quasi Gelfand triple.

The conjecture is motivated by the paper's generalization of ordinary Gelfand triples, replacing continuous embeddings by closedness. The authors state that it seems to be true, but report that attempts to prove it have failed; the weak conjecture is therefore open.