Thin-position conjecture for the knots $K_{2,1,3,7}$ and $K_{4,1,3,3}$

Let $B_i$ be the braids occurring in the paper's construction, and let $K_{2,1,3,7}$ and $K_{4,1,3,3}$ denote the corresponding knots. A thin position is an embedding realizing minimum width; minimum trunk and bridge number are defined analogously. Thin-position conjecture. For sufficiently complicated braids $B_i$, thin position of $K_{2,1,3,7}$ does not realize minimum trunk or minimum bridge number, while thin position of $K_{4,1,3,3}$ realizes minimum bridge number but not minimum trunk. These claims arise from embeddings whose thinness cannot currently be verified for the stated examples, so they remain conjectural.