Existence of a strongly homotopy-ribbon concordance that is not ribbon

Ascertain whether there exists a strongly homotopy-ribbon concordance between knots in S^3 that is not ribbon; equivalently, determine whether the class of strongly homotopy-ribbon concordances strictly contains the class of ribbon concordances.

Background

The paper recalls that ribbon concordances are strongly homotopy-ribbon, and this inclusion is central to their use of Agol’s partial order result. However, the potential strictness of this inclusion is a classical open problem in concordance theory.

Resolving whether there is a strongly homotopy-ribbon concordance that is not ribbon would clarify the relationship between these two classes and impact the structure of concordance partial orders and related obstructions used in the paper.