Strict triple-crossing lower bound for nonmonic Alexander polynomials

Let $K$ be a knot. Write $c_3(K)$ for its triple-crossing number and let $\Delta(K)$ denote its Alexander polynomial; write $\operatorname{breadth}(\Delta(K))$ for the breadth of this polynomial. Strict lower-bound conjecture. If $\Delta(K)$ is not monic, then

$$c_3(K)>\operatorname{breadth}(\Delta(K)).$$

The paper motivates this conjecture using the known bound $c_3(K)\geq 2g_c(K)$, where $g_c(K)$ is the canonical genus, together with $2g_c(K)\geq \operatorname{breadth}(\Delta(K))$. The proposed strict inequality is based on the authors' experiments and is not resolved in the supplied text.