Existence of a knot with distinct minimum and maximum thin-position heights

Let $K$ be a knot. For a thin position $\gamma$ of $K$, let $ht(\gamma)$ be the number of thick level spheres, and define $ht(K)$ and $ht_{\min}(K)$ to be the maximum and minimum, respectively, of $ht(\gamma)$ over all thin positions of $K$.

Height variation conjecture. There is a knot $K$ such that

$$ht_{\min}(K)<ht(K).$$

This conjecture asks whether a knot can have thin positions with different numbers of thick level spheres. The source provides no resolution, so the problem remains open.