Kauffman-polynomial span bound in terms of Turaev genus

Let $L$ be a link with crossing number $c(L)$ and Turaev genus $g_T(L)$, and let $D$ be a diagram of $L$. Write $F_L(a,z)$ for the Kauffman polynomial of $L$ and $\Lambda_D(a,z)$ for the corresponding Kauffman polynomial of the diagram $D$.

Kauffman-polynomial span conjecture. The spans satisfy

$$\operatorname{span} F_L(a,z)=\operatorname{span}\Lambda_D(a,z)\leq c(L)-2g_T(L).$$

The paper identifies this as a strengthening of a proposition needed to generalize its proof of the Turaev-genus-one characterization. Its status is not resolved in the supplied text.