Rigidity conjecture for hosting sets of prime knots

Let $K$) and $L$ be prime knots, and let $S(K)$ and $S(L)$ denote their hosting sets. Hosting-set rigidity conjecture. If

$$S(K)=S(L),$$

then $K=L$. This asks whether a prime knot is determined by the set of knot types represented by simple closed curves on a minimal genus Seifert surface of it; the broader problem for arbitrary knots is posed in the paper.