Existence of pseudo-meridians for all nontrivial knots

Establish whether every nontrivial knot K in S^3 admits a pseudo-meridian, namely an element of the knot group G(K) that normally generates G(K) but is not in the automorphic orbit of any meridian.

Background

A pseudo-meridian is a nontrivial element of the knot group that normally generates the group but is not an automorphic image of a meridian. The paper observes that any pseudo-meridian would be a persistent element, since if it became trivial under a Dehn filling then so would the meridian, contradicting Property P. Despite this connection and the abundance of persistent elements established in the paper, the existence of pseudo-meridians for all knots remains unsettled.

The conjecture is classical (attributed to Silver–Whitten–Williams) and is explicitly stated here as widely open, highlighting a gap between the demonstrated ubiquity of persistent elements and the still-unresolved status of pseudo-meridians.