Simultaneous crossing-and-wrapping diagram conjecture for solid torus links

Let $L$ be a solid torus link with crossing number $c$ and wrapping number $\operatorname{wr}$. A diagram of $L$ is said to realize both invariants when it has exactly $c$ crossings and $\operatorname{wr}$ passings. Simultaneous realization conjecture. There is a diagram $D$ of $L$ which has $c$ crossings and $\operatorname{wr}$ passings.

The authors observe this property for every link in their tabulation and state that its general validity seems reasonable, but no general proof is given.