Persistent free subgroup of rank two in hyperbolic knot groups

Determine whether the knot group G(K) of a hyperbolic knot K in S^3 contains a persistent free subgroup of rank two, meaning a free subgroup F_2 whose nontrivial elements remain nontrivial under all nontrivial Dehn fillings on K.

Background

The paper proves that cyclic persistent subgroups exist precisely for knots with no finite surgery and shows that most satellite knots admit persistent free subgroups of rank two. However, for hyperbolic knots, the existence of such non-cyclic persistent subgroups remains unsettled.

This question seeks to bridge that gap by asking whether hyperbolic knot groups contain a persistent free subgroup of rank two, reflecting a stronger form of persistence than the cyclic case.

References

On the other hand, to the best of our knowledge, no such examples are known for hyperbolic knots. Let K be a hyperbolic knot. Does G(K) admit a persistent free group of rank two?

— Dehn filling and the knot group II: Ubiquity of persistent elements  (2604.01697 - Ito et al., 2 Apr 2026) in Section 7.1: Further questions — Persistent subgroup (Question: non-cyclic persistent subgroup)