Zupan's width conjecture for satellite knots

Let $K$ be a satellite knot with companion $J$ and pattern $\hat{K}$ having wrapping number $n$. The width of a knot, denoted by $w(\cdot)$, is the knot width in the sense of Gabai. Zupan's width conjecture.

If $K$ is a satellite knot with companion $J$ and pattern $\hat{K}$ with wrapping number $n$, then

$$w(K) \geq n^2 \cdot w(J).$$

This conjecture is the analogue for knot width of the bridge-number inequality of Schubert and Schultens. The paper proves the corresponding equality $w(K)=q^2w(J)$ for a $(p,q)$-cable, but the general satellite-knot inequality stated here is not resolved in the supplied text.