Positive stable crossing growth conjecture for twist families

Let $\{K_n\}$ be a twist family of knots with wrapping number $\eta>1$. Define the lower stable crossing number by

$$\underline{c_s}(K_n)=\varliminf_{n\to\infty}\frac{c(K_n)}{n}.$$

Positive stable crossing growth conjecture.

$$\underline{c_s}(K_n)>0.$$

This weaker conjecture asks only for linear lower growth, rather than the exact slope predicted by the stable crossing number conjecture.