Positive stable braid index conjecture for non-coherent twist families

Let $\{K_n\}$ be a twist family of knots. Call the twist family non-coherent when its twisting strands are not coherently oriented. Define the lower stable braid index by

$$\underline{b_s}(K_n)=\varliminf_{n\to\infty}\frac{b(K_n)}{n}.$$

Positive stable braid index conjecture. If $\{K_n\}$ is non-coherent, then

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

This is presented as a weaker version of the stable braid index conjecture, and would provide a positive braid-index contribution for non-coherent twist families.