Width-minimizing positions that fail to minimize trunk or bridge number

Let $K$ and $K'$ be knots, with embeddings $k$ and $k'$, respectively. An embedding is width-minimizing when it realizes the knot's minimum width; similarly, trunk-minimizing and bridge-minimizing embeddings realize the minimum trunk and bridge number. Proposed counterexample conjecture. There exists a knot $K$ with an embedding $k$ that minimizes width but minimizes neither trunk nor bridge number, and there exists a knot $K'$ with an embedding $k'$ that minimizes both width and bridge number but does not minimize trunk. The paper gives concrete examples likely to satisfy this claim, but explicitly stops short of proving that the relevant embeddings minimize width.