The straight-number gap conjecture for alternating knots

Let $K$ be an alternating knot, and let $c(K)$ denote its crossing number and $\mathtt{str}(K)$ its straight number. Suppose

$$\mathtt{str}(K)=c(K)+n,$$

where $n\geq 1$, and let $K'$ be the knot obtained by increasing the number of full twists in some twist region. Straight-number gap conjecture. Then

$$\mathtt{str}(K')\geq c(K')+n.$$

The conjecture predicts that increasing full twists in a twist region does not decrease the difference between straight number and crossing number. The paper gives the example of $9_{32}$ and notes that increasing a twist produces $11_{91}$, for which the gap is larger, but no general proof is provided.