The coil number conjecture for alternating knots

Let $K$ be a knot. If $F$ is a closed surface containing $K$, define the coil number of $(F,K)$ by $r(F,K)=\min_{D\in\mathcal{D}_F}|\partial D\cap K|$, where $\mathcal{D}_F$ is the set of compressing disks for $F$, and define

$$r(K)=\max_{F\in\mathcal{F}}r(F,K),$$

where $\mathcal{F}$ is the set of closed surfaces containing $K$.

Alternating-knot coil number conjecture. For an alternating knot $K$, it holds that

$$r(K)=2.$$

The supplied text does not state a resolution of this claim.