Linear upper bound for the ribbonlength of ribbon knots

Let $K$ be a ribbon knot, let $\operatorname{Rib}(K)$ denote its minimal ribbonlength, and let $c(K)$ denote its crossing number. Linear ribbonlength conjecture. The minimal ribbonlength satisfies

$$\operatorname{Rib}(K) \leq C\cdot c(K) \quad \text{for some constant } C>0.$$

Such a bound would constrain the growth of minimal ribbonlength in terms of crossing number and clarify the relationship between ribbonlength-minimising configurations and crossing-minimising diagrams. The source presents this as a conjectural common assumption; its status is not established here.