Properness of the containment of ribbon 2-knot groups

Determine whether the containment of the class of ribbon 2-knot groups in the class \(\mathcal{G}(\mathcal{B})\) is proper, equivalently, whether there exists a group in \(\mathcal{G}(\mathcal{B})\) that is not a ribbon 2-knot group.

Background

The paper defines $\mathcal{G}(\mathcal{B})$ as a class of deficiency-one, weight-one groups with infinite cyclic abelianization obtained by composing a deficiency-one presentation of $\mathbb{Z}$ with a balanced presentation of the trivial group. It proves that this class lies between the classical knot groups and the 2-knot groups, and later observes that it contains all ribbon 2-knot groups.

The authors explain that deciding whether this containment is proper can be reduced to an Andrews–Curtis equivalence problem. Specifically, adjoining the meridian relator to a presentation in $\mathcal{B}$ gives a balanced presentation of the trivial group; the original group is a ribbon 2-knot group precisely when this balanced presentation is Andrews–Curtis equivalent to the standard trivial presentation.