Kauffman's Strong Conjecture for genus-one Seifert surfaces

Let $K$ be a slice knot, and let $F$ be a genus-one Seifert surface for $K$. For a simple closed curve $d$ on $F$, write $d^+$ for its positive push-off and $obreak\text{lk}(d,d^+)$ for its self-linking. Kauffman's Strong Conjecture. There \exists an essential simple closed curve $d$ on $F$ such that

$$\operatorname{lk}(d,d^+)=0$$

and $d$ is a slice knot. The paper's counterexamples to Kauffman's conjectures show that this assertion fails in general.