The h-ribbon characterization conjecture for ribbon groups

Let $G$ be a ribbon group for which topological surgery works. A knot $K$ is h-ribbon with group $G$ if there exists an epimorphism

$$\varphi\colon \pi_1(M_K)\twoheadrightarrow G$$

such that condition (Ext) from Theorem~ holds.

H-ribbon characterization conjecture. A knot $K$ is h-ribbon with group $G$ if and only if there exists an epimorphism $\varphi\colon \pi_1(M_K)\twoheadrightarrow G$ such that condition (Ext) from Theorem~ holds.

This proposes a converse to the criterion established earlier in the paper, under the assumptions that $G$ is a ribbon group and topological surgery works. The conjecture is presented as an open generalization of the main theorem.